AdamW has become one of the most effective optimizers for training large-scale models. We have also observed its effectiveness in the context of federated learning (FL). However, directly applying AdamW in federated learning settings poses significant challenges: (1) due to data heterogeneity, AdamW often yields high variance in the second-moment estimate v; (2) the local overfitting of AdamW may cause client drift; and (3) Reinitializing moment estimates (v, m) at each round slows down convergence. To address these challenges, we propose the first Federated AdamW algorithm, called FedAdamW, for training and fine-tuning various large models. FedAdamW aligns local updates with the global update using both a local correction mechanism and decoupled weight decay to mitigate local overfitting. FedAdamW efficiently aggregates the mean of the second-moment estimates to reduce their variance and reinitialize them. Theoretically, we prove that FedAdamW achieves a linear speedup convergence rate of O（p（L∆σ2l ）/（SKRε2） + （L∆）/R） without heterogeneity assumption, where S is the number of participating clients per round, K is the number of local iterations, and R is the total number of communication rounds. We also employ PAC-Bayesian generalization analysis to explain the effectiveness of decoupled weight decay in local training. Empirically, we validate the effectiveness of FedAdamW on language and vision Transformer models. Compared to several baselines, FedAdamW significantly reduces communication rounds and improves test accuracy.

Uplift modeling has obtained significant attention, with broad applications in medicine, economics, and marketing. For example, in a push notification scenario, accurately estimating the uplift of different push frequencies on user activation and notification switch close rate is critical for balancing user experience and business goals. Existing methods only use binary labels, i.e., convert or not within the observational window. However, they ignore time information (e.g., users who convert on day 1 vs. day 14 reflect different sensitivities) and fail to model potential closures outside the window, i.e., due to treatments always taking time to manifest causal impacts on outcomes, the potential outcomes of interest cannot be observed promptly and accurately. Failing to account for these issues can result in skewed uplift modeling. To address this gap, this work examines how observation timing influences the assessment of uplift by explicitly modeling the potential response time. Theoretical analysis establishes the conditions for identifiability under delayed feedback scenarios. We introduce CFR-DF (Counterfactual Regression with Delayed Feedback), a systematic framework that jointly learns both the latent response times and the underlying potential outcomes. Empirical evaluations on synthetic and real-world datasets, including an A/B test with over 1 billion users for 14 days, validate the approach, demonstrating its ability to handle temporal delays and improve estimation accuracy compared to previous uplift modeling methods.

In marketing, optimizing subsidy allocation to maximize overall profits is of substantial economic importance. Prior research has employed treatment effect estimation techniques to identify subsidy-sensitive items and design corresponding allocation strategies. However, more accurate treatment effect estimations do not necessarily lead to better allocations, underscoring the critical influence of decision boundaries in decision-making. This paper argues that optimal allocation fundamentally depends on predicting the expected optimal subsidy, a challenge distinct from conventional treatment effect estimation or causal decision-making, which existing approaches fail to address. To fill this gap, we introduce a two-stage Counterfactual optimal subsidy Learning method with an Asymmetric reward (CoLA). In the first stage, we derive a coarse estimate of the expected subsidy threshold by exploiting order information and the conditional independence between expected and observed subsidies. In the second stage, we refine these estimates using an asymmetric loss function, leading to more robust predictions. Under practical budget constraints, we prioritize candidates based on their Sharpe ratios to determine the final subsidy allocation strategy. Experiments on three public datasets and an online A/B test show that our method achieves significant performance improvements, yielding the highest total profit and incremental leverage ratios.

Post-click conversion rate (CVR) prediction is a central task in recommender systems, yet selection bias creates a severe distributional gap between the clicked training samples and the entire inference space. To address selection bias, propensity-based methods such as inverse propensity scoring (IPS) and doubly robust (DR) have been adopted, which aim to estimate the unbiased learning objective from biased training samples. However, these approaches assume strictly positive propensities, implying every user-item pair has a nonzero probability of interaction. In practice, such positivity assumption maybe violated, for example, in food-delivery platforms, some restaurants located more than 10 kilometers away will be blocked for recommendation. In this study, we theoretically show that when such zero-propensity samples, termed extrapolation samples exist, both IPS and DR estimators become biased. To overcome this limitation, we propose ExtraDebias method, which enables debiased recommendation in both non-extrapolation and extrapolation samples. Specifically, we first train a propensity model to identify extrapolation samples with extremely small propensity estimates, then estimate their pseudo-label intervals, and derive an upper bound of the learning objective for extrapolation samples. By minimizing the derived upper bound, debiased learning on extrapolation samples is ensured, while unbiased learning on non-extrapolation samples is achieved by standard IPS. Experiments on four real-world offline datasets and one online A/B test show that ExtraDebias effectively minimizes prediction errors on extrapolation samples and achieves optimal performance.

In Transformers, Position Embeddings (PEs) significantly influence Length Generalization (LG) performance, yet their fundamental role remains unclear. In this work, we identify two fundamental limitations of PEs in achieving LG: the inability to acquire new operators beyond training data, and the inability to deal with inconsistent computational roles of the same positions across scales. To formalize these limitations, we introduce computational representation complexity which measures the number of distinct unit operators required for a task, and the notion of canonicality, which characterizes operator-position consistency as the scale varies. We prove that PEs cannot achieve LG for mappings whose computational representation complexity increases with scale, nor for mappings that are non-canonical. We further show that PEs enable LG when these two failure modes are absent. In this case, LG can be achieved with a PE whose positional relation function characterizes the computational representation. To enhance LG for non-canonical mappings, we introduce scale hint, allowing flexible instance scaling. We also propose a learning-based position embedding framework that automatically learns positional relations. Our work provides theoretical insights and practical strategies for improving LG in Transformers.

Skip connection is an essential ingredient for modern deep models to be deeper and more powerful. Despite their huge success in normal scenarios (state-of-the-art classification performance on natural examples), we investigate and identify an interesting property of skip connections under adversarial scenarios, namely, the use of skip connections allows easier generation of highly transferable adversarial examples. Specifically, in ResNet-like models (with skip connections), we find that biasing backpropagation to favor gradients from skip connections-while suppressing those from residual modules via a decay factor-allows one to craft adversarial examples with high transferability. Based on this insight, we propose the Skip Gradient Method (SGM). Although starting from ResNet-like models in vision domains, we further extend SGM to more advanced architectures, including Vision Transformers (ViTs), models with varying-length paths, and other domains such as natural language processing. We conduct comprehensive transfer-based attacks against diverse model families, including ResNets, Transformers, Inceptions, Neural Architecture Search-based models, and Large Language Models (LLMs). The results demonstrate that employing SGM can greatly improve the transferability of crafted attacks in almost all cases. Furthermore, we demonstrate that SGM can still be effective under more challenging settings such as ensemble-based attacks, targeted attacks, and against defense equipped models. At last, we provide theoretical explanations and empirical insights on how SGM works. Our findings not only motivate new adversarial research into the architectural characteristics of models but also open up further challenges for secure model architecture design.

The recent advancement in video temporal grounding (VTG) has significantly enhanced fine-grained video understanding, primarily driven by multimodal large language models (MLLMs). With superior multimodal comprehension and reasoning abilities, VTG approaches based on MLLMs (VTG-MLLMs) are gradually surpassing traditional fine-tuned methods. They not only achieve competitive performance but also excel in generalization across zero-shot, multi-task, and multi-domain settings. Despite extensive surveys on general video-language understanding, comprehensive reviews specifically addressing VTG-MLLMs remain scarce. To fill this gap, this survey systematically examines current research on VTG-MLLMs through a three-dimensional taxonomy: 1) the functional roles of MLLMs, highlighting their architectural significance; 2) training paradigms, analyzing strategies for temporal reasoning and task adaptation; and 3) video feature processing techniques, which determine spatiotemporal representation effectiveness. We further discuss benchmark datasets, evaluation protocols, and summarize empirical findings. Finally, we identify existing limitations and propose promising research directions.

In response to the challenges posed by the extensive parameter updates required for full fine-tuning of large-scale pre-trained models, parameter-efficient fine-tuning (PEFT) methods, exemplified by Low-Rank Adaptation (LoRA), have emerged. LoRA simplifies the fine-tuning process but may still struggle with a certain level of redundancy in low-rank matrices and limited effectiveness from merely increasing their rank. To address these issues, a natural idea is to enhance the independence and diversity of the learning process for the low-rank matrices. Therefore, we propose Masked LoRA Experts (MLAE), an innovative approach that applies the concept of masking to visual PEFT. Our method incorporates a cellular decomposition strategy that treats rank-1 components as experts defined under the chosen LoRA parameterization, thus enhancing diversity among update components. Additionally, we introduce a binary mask matrix that selectively activates these experts during training to promote more diverse and anisotropic learning, based on expert-level dropout strategies. Our investigations reveal that this selective activation not only enhances performance but also fosters a more diverse acquisition of knowledge with a marked decrease in parameter similarity among MLAE, significantly boosting the quality of the model. Remarkably, MLAE achieves new state-of-the-art (SOTA) performance with an average accuracy score of 78.8% on the VTAB-1k benchmark and 90.9% on the FGVC benchmark, surpassing the previous SOTA result by an average of 0.8% on both benchmarks. Moreover, MLAE shows strong generalization across diverse tasks, including LLM fine-tuning, semantic segmentation, and image/video-text understanding, underscoring its versatility and effectiveness in advancing PEFT.

Video generation requires modeling a vast spatiotemporal space, which demands significant computational resources and data usage. To reduce the complexity, the prevailing approaches employ a cascaded architecture to avoid direct training with full resolution latent. Despite reducing computational demands, the separate optimization of each sub-stage hinders knowledge sharing and sacrifices flexibility. This work introduces a unified pyramidal flow matching algorithm. It reinterprets the original denoising trajectory as a series of pyramid stages, where only the final stage operates at the full resolution, thereby enabling more efficient video generative modeling. Through our sophisticated design, the flows of different pyramid stages can be interlinked to maintain continuity. Moreover, we craft autoregressive video generation with a temporal pyramid to compress the full-resolution history. The entire framework can be optimized in an end-to-end manner and with a single unified Diffusion Transformer (DiT). Extensive experiments demonstrate that our method supports generating high-quality 5-second (up to 10-second) videos at 768p resolution and 24 FPS within 20.7k A100 GPU training hours. All code and models are open-sourced at https://pyramid-flow.github.io.

In the field of equivariant networks, achieving affine equivariance, particularly for general group representations, has long been a challenge. In this paper, we propose the steerable EquivarLayer, a generalization of InvarLayer (Li et al., 2024), by building on the concept of equivariants beyond invariants. The steerable EquivarLayer supports affine equivariance with arbitrary input and output representations, marking the first model to incorporate steerability into networks for the affine group. To integrate it with canonicalization, a promising approach for making pre-trained models equivariant, we introduce a novel Det-Pooling module, expanding the applicability of EquivarLayer and the range of groups suitable for canonicalization. We conduct experiments on image classification tasks involving group transformations to validate the steerable EquivarLayer in the role of a canonicalization function, demonstrating its effectiveness over data augmentation.

The Mixture of Experts (MoE) architecture has emerged as a promising solution to reduce computational overhead by selectively activating subsets of model parameters. The effectiveness of MoE models depends primarily on their routing mechanisms, with the widely adopted Top-K routing scheme used for activating experts. However, the Top-K scheme has notable limitations, including unnecessary activations and underutilization of experts. In this work, rather than modifying the routing mechanism as done in previous studies, we propose the Ternary Choice MoE (TC-MoE), a novel approach that expands the expert space by applying the ternary set {-1, 0, 1} to each expert. This expansion allows more efficient and effective expert activations without incurring significant computational costs. Additionally, given the unique characteristics of the expanded expert space, we introduce a new load balance loss and reward loss to ensure workload balance and achieve a flexible trade-off between effectiveness and efficiency. Extensive experiments demonstrate that TC-MoE achieves an average improvement of over 1.1% compared with traditional approaches, while reducing the average number of activated experts by up to 9%. These results confirm that TC-MoE effectively addresses the inefficiencies of conventional routing schemes, offering a more efficient and scalable solution for MoE-based large language models. Code and models are available at https://github.com/stiger1000/TC-MoE.

Large language models (LLMs) can solve an increasing number of complex reasoning tasks while making surprising mistakes in basic numerical understanding and processing (such as $9.11 > 9.9$). The latter ability is essential for tackling complex arithmetic and mathematical problems and serves as a foundation for most reasoning tasks, but previous work paid little attention to it or only discussed several restricted tasks (like integer addition). In this paper, we comprehensively investigate the numerical understanding and processing ability (NUPA) of LLMs. Firstly, we introduce a benchmark covering four common numerical representations and 17 distinct numerical tasks in four major categories, resulting in 41 meaningful combinations in total. These tasks are derived from primary and secondary education curricula, encompassing nearly all everyday numerical understanding and processing scenarios, and the rules of these tasks are very simple and clear. Through the benchmark, we find that current LLMs fail frequently in many of the tasks. To study the problem, we train small models with existing and potential techniques for enhancing NUPA (such as tokenizers, PEs, and number formats), comprehensively evaluating their effectiveness using our testbed. We also finetune practical-scale LLMs on our proposed NUPA tasks and find that 1) naive finetuning can improve NUPA a lot on many but not all tasks, and 2) surprisingly, techniques designed to enhance NUPA prove ineffective for finetuning pretrained models. We further explore the impact of chain-of-thought techniques on NUPA. Our work provides a more detailed and comprehensive understanding of NUPA in LLMs.

Training Large Language Models (LLMs) presents a significant communication bottleneck, predominantly due to the growing scale of the gradient to communicate across multi-device clusters. However, how to mitigate communication overhead in practice remains a formidable challenge due to the weakness of the methodology of the existing compression methods, especially the neglect of the characteristics of the gradient. In this paper, we consider and demonstrate the low-rank properties of gradient and Hessian observed in LLMs training dynamic, and take advantage of such natural properties to design SEPARATE, a simple low-rank projection for gradient compression in modern large-scale model training processes. SEPARATE realizes dimensional reduction by common random Gaussian variables and an improved moving average error-feedback technique. We theoretically demonstrate that SEPARATE-based optimizers maintain the original convergence rate for SGD and Adam-Type optimizers for general non-convex objectives. Experimental results show that SEPARATE accelerates training speed by up to 2× for GPT-2-Medium pre-training, and improves performance on various benchmarks for LLAMA2-7B fine-tuning.

Implicit feedback recommendation is challenged by the missing negative feedback essential for effective model training. Existing methods often resort to negative sampling, a technique that assumes unlabeled interactions as negative samples. This assumption risks misclassifying potential positive samples within the unlabeled data, thereby undermining model performance. To address this issue, we introduce PURL, a model-agnostic framework that reframes implicit feedback recommendation as a weakly supervised learning task, eliminating the need for negative samples. However, its unbiasedness hinges on the accurate estimation of the class prior. To address this challenge, we propose Progressive Proximal Transport (PPT), which estimates the class prior by minimizing the proximal transport cost between positive and unlabeled samples. Experiments on three real-world datasets validate the efficacy of PURL in terms of improved recommendation quality. Code is available at https://github.com/HowardZJU/weakrec.

Low-Dimension-to-High-Dimension (LDHD) generalization, a subset of Out-of-Distribution (OOD) generalization, involves training on a low-dimensional subspace and testing in a high-dimensional space. Assuming instances are generated from latent variables reflecting problem scale, LDHD generalization captures the inherent scaling challenge of length generalization. We theoretically show that LDHD generalization is unattainable without appropriate inductive bias. Focusing on Boolean functions, we demonstrate that different architectures trained with (S)GD converge to min-degree interpolators w.r.t. different linearly independent sets, achieving LDHD generalization only when the target function aligns with this bias. From the perspective of LDHD generalization for length generalization, we explain the success of CoT in restructuring latent space for improved LDHD generalization. We further propose a principle for designing position embeddings to address both LDHD generalization and data format nuisances separately. Following the principle, we introduce RPE-Square, a novel embedding that enhances RPE to better handle data formats.

Kolmogorov-Arnold Networks (KANs) have seen great success in scientific domains thanks to spline activation functions, becoming an alternative to Multi-Layer Perceptrons (MLPs). However, spline functions may not respect symmetry in tasks, which is crucial prior knowledge in machine learning. In this paper, we propose Equivariant Kolmogorov-Arnold Networks (EKAN), a method for incorporating arbitrary matrix group equivariance into KANs, aiming to broaden their applicability to more fields. We first construct gated spline basis functions, which form the EKAN layer together with equivariant linear weights, and then define a lift layer to align the input space of EKAN with the feature space of the dataset, thereby building the entire EKAN architecture. Compared with baseline models, EKAN achieves higher accuracy with smaller datasets or fewer parameters on symmetry-related tasks, such as particle scattering and the three-body problem, often reducing test MSE by several orders of magnitude. Even in non-symbolic formula scenarios, such as top quark tagging with three jet constituents, EKAN achieves comparable results with state-of-the-art equivariant architectures using fewer than $40\%$ of the parameters, while KANs do not outperform MLPs as expected. Code and data are available at [https://github.com/hulx2002/EKAN](https://github.com/hulx2002/EKAN).

Symmetry is widely applied in problems such as the design of equivariant networks and the discovery of governing equations, but in complex scenarios, it is not known in advance. Most previous symmetry discovery methods are limited to linear symmetries, and recent attempts to discover nonlinear symmetries fail to explicitly get the Lie algebra subspace. In this paper, we propose LieNLSD, which is, to our knowledge, the first method capable of determining the number of infinitesimal generators with nonlinear terms and their explicit expressions. We specify a function library for the infinitesimal group action and aim to solve for its coefficient matrix, proving that its prolongation formula for differential equations, which governs dynamic data, is also linear with respect to the coefficient matrix. By substituting the central differences of the data and the Jacobian matrix of the trained neural network into the infinitesimal criterion, we get a system of linear equations for the coefficient matrix, which can then be solved using SVD. On top quark tagging and a series of dynamic systems, LieNLSD shows qualitative advantages over existing methods and improves the long rollout accuracy of neural PDE solvers by over $20\%$ while applying to guide data augmentation. Code and data are available at [https://github.com/hulx2002/LieNLSD](https://github.com/hulx2002/LieNLSD).

Designing experiments for causal effect estimation remains an enduring topic in both machine learning and statistics. While much of the existing statistical literature focuses on using central limit theorems to analyze asymptotic properties of estimators, a parallel line of research has emerged around theoretical tools that provide finite-sample error bounds, offering performance on par with—or superior to—the asymptotic approaches. These finite-sample results are especially relevant in active sampling settings where the sample size is limited (for instance, under privacy or cost constraints). In this paper, we develop a finite-sample estimator with sample complexity analysis and extend its applicability to social networks. Through simulations and real-world experiments, we show that our method achieves higher estimation accuracy with fewer samples than traditional estimators endowed with asymptotic normality and other estimators backed by finite-sample guarantees.

Large language models (LLMs) have achieved remarkable successes on various tasks. However, recent studies have found that there are still significant challenges to the logical reasoning abilities of LLMs, which can be categorized into the following two aspects: (1) Logical question answering: LLMs often fail to generate the correct answer within a complex logical problem which requires sophisticated deductive, inductive or abductive reasoning given a collection of premises and constrains. (2) Logical consistency: LLMs are prone to producing responses contradicting themselves across different questions. For example, a state-of-the-art question-answering LLM Macaw, answers Yes to both questions Is a magpie a bird? and Does a bird have wings? but answers No to Does a magpie have wings?. To facilitate this research direction, we comprehensively investigate the most cutting-edge methods and propose a detailed taxonomy. Specifically, to accurately answer complex logic questions, previous methods can be categorized based on reliance on external solvers, prompts, and fine-tuning. To avoid logical contradictions, we discuss concepts and solutions of various logical consistencies, including implication, negation, transitivity, factuality consistencies, and their composites. In addition, we review commonly used benchmark datasets and evaluation metrics, and discuss promising research directions, such as extending to modal logic to account for uncertainty and developing efficient algorithms that simultaneously satisfy multiple logical consistencies.

Heterogeneous treatment effect (HTE) estimation from observational data poses significant challenges due to treatment selection bias. Existing methods address this bias by minimizing distribution discrepancies between treatment groups in latent space, focusing on global alignment. However, the fruitful aspect of local proximity, where similar units exhibit similar outcomes, is often overlooked. In this study, we propose Proximity-enhanced CounterFactual Regression (CFR-Pro) to exploit proximity for enhancing representation balancing within the HTE estimation context. Specifically, we introduce a pair-wise proximity regularizer based on optimal transport to incorporate the local proximity in discrepancy calculation. However, the curse of dimensionality renders the proximity measure and discrepancy estimation ineffective-exacerbated by limited data availability for HTE estimation. To handle this problem, we further develop an informative subspace projector, which trades off minimal distance precision for improved sample complexity. Extensive experiments demonstrate that CFR-Pro accurately matches units across different treatment groups, effectively mitigates treatment selection bias, and significantly outperforms competitors. Code is available at https://github.com/HowardZJU/CFR-Pro.

Equivariant networks enhance model efficiency and generalization by embedding symmetry priors into their architectures. However, most existing methods, primarily based on group convolutions and steerable convolutions, face significant limitations when dealing with complex transformation groups, particularly the projective group, which plays a crucial role in vision. In this work, we tackle the challenge by constructing projective equivariant networks based on differential invariants. Using the moving frame method with a carefully selected cross section tailored for multi-dimensional functions, we derive a complete and concise set of second-order fundamental differential invariants of the projective group. We provide a rigorous analysis of the properties and transformation relationships of their underlying components, yielding a further simplified and unified set of fundamental differential invariants, which facilitates both theoretical analysis and practical applications. Building on this foundation, we develop PDINet, the first framework for deep projective equivariant networks, achieving full projective equivariance without discretizing or sampling the group. Empirical results on the projectively transformed STL-10 and Imagenette datasets show that PDINet achieves improvements of 11.39\% and 5.66\% in accuracy over the respective standard baselines under out-of-distribution settings, demonstrating its strong generalization to complex geometric transformations.

Multimodal Large Language Models (MLLMs) have achieved considerable accuracy in Optical Character Recognition (OCR) from static images. However, their efficacy in video OCR is significantly diminished due to factors such as motion blur, temporal variations, and visual effects inherent in video content. To provide clearer guidance for training practical MLLMs, we introduce MME-VideoOCR benchmark, which encompasses a comprehensive range of video OCR application scenarios. MME-VideoOCR features 10 task categories comprising 25 individual tasks and spans 44 diverse scenarios. These tasks extend beyond text recognition to incorporate deeper comprehension and reasoning of textual content within videos. The benchmark consists of 1,464 videos with varying resolutions, aspect ratios, and durations, along with 2,000 meticulously curated, manually annotated question-answer pairs. We evaluate 18 state-of-the-art MLLMs on MME-VideoOCR, revealing that even the best-performing model (Gemini-2.5 Pro) achieves only an accuracy of 73.7%. Fine-grained analysis indicates that while existing MLLMs demonstrate strong performance on tasks where relevant texts are contained within a single or few frames, they exhibit limited capability in effectively handling tasks that demand holistic video comprehension. These limitations are especially evident in scenarios that require spatio-temporal reasoning, cross-frame information integration, or resistance to language prior bias. Our findings also highlight the importance of high-resolution visual input and sufficient temporal coverage for reliable OCR in dynamic video scenarios.

The expanding computational costs and limited resources underscore the critical need for budgeted-iteration training, which aims to achieve optimal learning within predetermined iteration budgets. While learning rate schedules fundamentally govern the performance of different networks and tasks, particularly in budgeted-iteration scenarios, their design remains largely heuristic, lacking theoretical foundations. In addition, the optimal learning rate schedule requires extensive trial-and-error selection, making the training process inefficient. In this work, we propose the Unified Budget-Aware (UBA) schedule, a theoretically grounded learning rate schedule that consistently outperforms commonly-used schedules among diverse architectures and tasks under different constrained training budgets. First, we bridge the gap by constructing a novel training budget-aware optimization framework, which explicitly accounts for the robustness to landscape curvature variations. From this framework, we derive the UBA schedule, controlled by a single hyper-parameter $\varphi$ that provides a trade-off between flexibility and simplicity, eliminating the need for per-network numerical optimization. Moreover, we establish a theoretical connection between $\varphi$ and the condition number, adding interpretation and justification to our approach. Besides, we prove the convergence for different values of $\varphi$. We offer practical guidelines for $\varphi$ selection via theoretical analysis and empirical results. Extensive experimental results show that UBA $\textit{consistently surpasses}$ the commonly-used schedules across diverse vision and language tasks, spanning network architectures (e.g., ResNet, OLMo) and scales, under different training-iteration budgets.

Training time-series forecasting models poses unique challenges in loss function design. Most existing approaches adopt temporal mean squared error, but this study reveals two critical limitations: (1) it ignores the presence of label autocorrelation, which biases it from the true label sequence likelihood; (2) it involves excessive number of tasks, which complicates optimization, especially for long-term forecasting. To address these issues, we introduce Time-o1, a transform-enhanced loss function for time-series forecasting. The central idea is to transform the label sequence into decorrelated components with discriminated significance. Models are then trained to align the most significant components, thereby effectively mitigating label autocorrelation and reducing task amount. Experiments demonstrate that Time-o1 achieves state-of-the-art performance and is compatible with various forecast models. Code is available at https://github.com/Master-PLC/Time-o1.

Iterative imputation is a prevalent method for missing data imputation, where each feature is imputed iteratively by treating it as a target variable estimated from all other features. However, iterative imputation method suffers from two principal limitations: (1) it imposes a single parametric model form to impute all features, neglecting the potential for optimal models to vary among features, which risks model misspecification; and (2) it assumes every feature contains missing values, overlooking the potential presence of non-missing features, termed as oracle features, which are informative for imputation. To address these limitations, we propose kernel point imputation (KPI), a bi-level optimization framework for iterative missing data imputation. At the inner level, KPI adaptively learns the optimal model form for each feature within a reproducing kernel Hilbert space, addressing limitation (1). At the outer level, KPI utilizes oracle features as supervisory signals to iteratively refine the imputations, addressing limitation (2). Experiments demonstrate that KPI outperforms competitive imputation methods. Code is available at https://github.com/FMLYD/kpi.git.

Transformer models have driven breakthroughs across various language tasks by their strong capability to learn rich contextual representations. Scaling them to improve representation, however, often demands substantial memory and compute costs, such as the Key-Value (KV) cache used during auto-regressive decoding. Skip connections offer a promising way to improve representation without bloating resource usage, yet most prior works either improve expressivity while leaving KV costs unchanged, or reduce memory at the cost of weaker representation. In this work, we propose SkipV1Former, a Transformer variant that uses skip connections from the first layer's Value heads to strengthen model representation and reduce KV cache. Specifically, from the second block onward, each layer reuses half of its Value heads from the very first layer, while computing the other half as usual-cutting Value projections and V cache by nearly 50 \%. Theoretically, we show that routing uncompressed first-layer Values into deeper layers restores information lost to compression and accelerates the model’s implicit mesa-optimization-a key pattern of Transformer in auto-regressive tasks. Empirically, across different model scales, SkipV1Former delivers consistent reductions of approximately 25 \% in KV cache while improving perplexity relative to standard Multi-Head Attention (MHA) Transformers and some advanced variants. Moreover, we propose a recipe for uptraining existing MHA Transformer checkpoints to SkipV1Former with only 10-15\% additional compute. Finally, SkipV1Former can seamlessly combine advanced methods like Group-Query Attention and Multi-Latent Attention to achieve further KV cache savings and performance improvement. When combined with YOCO, it cuts KV cache size by nearly 50 \% while still improving performance. The code is available at: https://github.com/Zhoutong-Wu/SkipV1Former.

Zeroth-order Optimization (ZO) has received wide attention in machine learning, especially when computing full gradient is expensive or even impossible. Recently, ZO has emerged as an important paradigm for memory-efficient fine-tuning of large language models (LLMs), circumventing the memory overhead of backpropagation. However, existing ZO gradient estimators exhibit dimension-dependent variance scaling as $\Theta(d)$, leading to dimension-dependent convergence rates without further assumptions on the objective function, which is prohibitive for large-scale LLM parameters. To address this problem, we present a Pseudo-Zeroth-Order (PseuZO) framework for optimizing composite objective functions, especially large-scale models: $ \min_{\mathbf{x} \in \mathcal{X}} \mathcal{F}(\mathbf{x})= \bbE_{\mathbf{z}} g\circ h(\mathbf{x};\mathbf{z}) $, where $h$ represents complex, high-dimensional representations and $g$ is a task-specific loss. While existing zeroth-order methods estimate gradients with final loss functions, our PseuZO algorithm estimate the Jacobian matrix of $h(\mathbf{x})$ with the model output $\mathbf{o}= h(\mathbf{x})$, and the gradient of the loss function on model output $\mathbf{e} = \nabla_{\mathbf{o}} g(\mathbf{o})$, and apply exponential moving average on Jacobian estimators to reduce the variance. Moreover, we use the sliding window technique to reduce memory costs. Our algorithm achieves an $O( \max \lbrace \alpha_1 L\epsilon^{-2}, \alpha_1 L \sigma_2^2\epsilon^{-4} \rbrace )$ convergence rate, where $\alpha_1$ is the effective dimension of $\mathcal{F}$. Experimental results demonstrate that PseuZO outperforms MeZO and MeZO-SVRG in classification, multiple choice and generation tasks in both full-parameter and PEFT fine-tuning settings by boosting convergence in the early stages of training. For instance, under the same computation time, with respect to SST2 task, PesuZO gets 9.8\% higher accuracy than MeZO (91.2\% v.s. 82.4\%). With the sliding window technique, our PseuZO achieves $70\%\sim80\%$ memory reduction compared to FO-SGD for different model sizes as PseuZO only introduced a small dimension-independent memory overhead, which enables efficient scaling of the model size. The code is available at https://github.com/YangBigMn/PseuZO. $\newcommand{\bbE}{\mathbb{E}}$

Conventional research on large language models (LLMs) has primarily focused on refining output distributions, while paying less attention to the decoding process that transforms these distributions into final responses. Recent advances, such as scaling the computation of inference time with reward models, have underscored the importance of decoding, but these methods often suffer from high computational costs and limited applicability. In this paper, we revisit LLM generation through the lens of recommender systems, conceptualizing the decoding process as analogous to the ranking stage in recommendation pipelines. From this perspective, we observe that both traditional decoding methods and reward models exhibit clear limitations such as redundancy. Motivated by this insight, we propose Language Ranker, a novel framework that introduces a lightweight module to rerank candidate responses using features extracted by the base model. Experiments across a wide range of tasks show that Language Ranker achieves performance comparable to large-scale reward models, while requiring only <0.5M additional parameters, significantly reducing the computational overhead during both training and inference stages. This highlights the efficiency and effectiveness of our method, showcasing its potential to fully unlock the capabilities of LLMs.

This paper introduces PaZO, a preconditioned accelerated zeroth-order optimization algorithm for fine-tuning large language models (LLMs). First, we theoretically demonstrate the necessity of preconditioning in zeroth-order optimization, proving that zeroth-order stochastic gradient descent (ZO-SGD) alone fails to achieve the ideal convergence rate. Building on this, we propose a Preconditioned Simultaneous Perturbation Stochastic Approximation (PSPSA) and theoretical version of PaZO, and demonstrate that setting the order of preconditioner as $-1/2$ in PSPSA yields the improved convergence rate for PaZO. Moreover, we design a practical version of PaZO that stabilizes training via diagonal Hessian estimate and moving average technique. Extensive experiments on diverse downstream tasks with models like RoBERTa-large and OPT show PaZO’s effectiveness. Compared to other zeroth-order baselines, PaZO achieves better performance across models and tasks.

In this paper, we propose AdaMSS, an adaptive multi-subspace approach for parameter-efficient fine-tuning of large models. Unlike traditional parameter-efficient fine-tuning methods that operate within a large single subspace of the network weights, AdaMSS leverages subspace segmentation to obtain multiple smaller subspaces and adaptively reduces the number of trainable parameters during training, ultimately updating only those associated with a small subset of subspaces most relevant to the target downstream task. By using the lowest-rank representation, AdaMSS achieves more compact expressiveness and finer tuning of the model parameters. Theoretical analyses demonstrate that AdaMSS has better generalization guarantee than LoRA, PiSSA, and other single-subspace low-rank-based methods. Extensive experiments across image classification, natural language understanding, and natural language generation tasks show that AdaMSS achieves comparable performance to full fine-tuning and outperforms other parameter-efficient fine-tuning methods in most cases, all while requiring fewer trainable parameters. Notably, on the ViT-Large model, AdaMSS achieves 4.7\% higher average accuracy than LoRA across seven tasks, using just 15.4\% of the trainable parameters. On RoBERTa-Large, AdaMSS outperforms PiSSA by 7\% in average accuracy across six tasks while reducing the number of trainable parameters by approximately 94.4\%. These results demonstrate the effectiveness of AdaMSS in parameter-efficient fine-tuning. The code for AdaMSS is available at https://github.com/jzheng20/AdaMSS.

Existing video hashing techniques for large-scale video retrieval often overlook inherent audio signals, which can potentially compromise retrieval performance. Incorporating both visual and audio signals, however, complicates neural architecture design, rendering the manual crafting of joint audio-visual neural network models challenging. To address this issue, we propose AV-NAS, a method that leverages data-driven Neural Architecture Search (NAS) within a tailored audio-visual network space to automatically discover the optimal video hashing network. Our approach offers: (1) a versatile multi-level semantic architecture based on audio-visual signals, defining a mixed search space encompassing diverse network modules such as MLP, CNN, Transformer, and Mamba, as well as operations like Add, Hadamard, SiLU, LayerNorm, and Skip; (2) a differentiable relaxation of the combinatorial search problem, converting it into a unified differentiable optimization problem which we tackle through our ''coarse search-pruning-finetuning'' strategy. Our experiments on large-scale video datasets show that AV-NAS can discover architectures distinct from expert designs and lead to substantial performance improvements over current state-of-the-art methods including the recently emerged AVHash.

Although adaptive gradient methods have been extensively used in deep learning, their convergence rates proved in the literature are all slower than that of SGD, particularly with respect to their dependence on the dimension. This paper considers the classical RMSProp and its momentum extension and establishes the convergence rate of $\frac{1}{T}\sum_{k=1}^TE\left[||\nabla f(\mathbf{x}^k)||_1\right]\leq O(\frac{\sqrt{d}C}{T^{1/4}})$ measured by $\ell_1$ norm without the bounded gradient assumption, where $d$ is the dimension of the optimization variable, $T$ is the iteration number, and $C$ is a constant identical to that appeared in the optimal convergence rate of SGD. Our convergence rate matches the lower bound with respect to all the coefficients except the dimension $d$. Since $||\mathbf{x}||_2\ll ||\mathbf{x}||_1\leq\sqrt{d}||\mathbf{x}||_2$ for problems with extremely large $d$, our convergence rate can be considered to be analogous to the $\frac{1}{T}\sum_{k=1}^TE\left[||\nabla f(\mathbf{x}^k)||_2\right]\leq O(\frac{C}{T^{1/4}})$ rate of SGD in the ideal case of $||\nabla f(\mathbf{x})||_1=\varTheta(\sqrt{d})||\nabla f(\mathbf{x})||_2$.

Irreducible Cartesian tensors (ICTs) play a crucial role in the design of equivariant graph neural networks, as well as in theoretical chemistry and chemical physics. Meanwhile, the design space of available linear operations on tensors that preserve symmetry presents a significant challenge. The ICT decomposition and a basis of this equivariant space are difficult to obtain for high-rank tensors. After decades of research, Bonvicini (2024) has recently achieved an explicit ICT decomposition for $n=5$ with factorial time/space complexity. In this work we, for the first time, obtain decomposition matrices for ICTs up to rank $n=9$ with reduced and affordable complexity, by constructing what we call path matrices. The path matrices are obtained via performing chain-like contractions with Clebsch-Gordan matrices following the parentage scheme. We prove and leverage that the concatenation of path matrices is an orthonormal change-of-basis matrix between the Cartesian tensor product space and the spherical direct sum spaces. Furthermore, we identify a complete orthogonal basis for the equivariant space, rather than a spanning set (Pearce-Crump, 2023b), through this path matrices technique. Our method avoids the RREF algorithm and maintains a fully analytical derivation of each ICT decomposition matrix, thereby significantly improving the algorithm’s speed to obtain arbitrary rank orthogonal ICT decomposition matrices and orthogonal equivariant bases. We further extend our result to the arbitrary tensor product and direct sum spaces, enabling free design between different spaces while keeping symmetry. The Python code is available at https://github.com/ShihaoShao-GH/ICT-decomposition-and-equivariant-bases, where the $n=6,\dots,9$ ICT decomposition matrices are obtained in 1s, 3s, 11s, and 4m32s on 28-core Intel Xeon Gold 6330 CPU @ 2.00GHz, respectively.

Recently, deep clustering methods have achieved remarkable results compared to traditional clustering approaches. However, its performance remains constrained by the absence of annotations. A thought-provoking observation is that there is still a significant gap between deep clustering and semi-supervised classification methods. Even with only a few labeled samples, the accuracy of semi-supervised learning is much higher than that of clustering. Given that we can annotate a small number of samples in a certain unsupervised way, the clustering task can be naturally transformed into a semi-supervised setting, thereby achieving comparable performance. Based on this intuition, we propose ClusMatch, a unified positive and negative pseudo-label learning based semi-supervised learning framework, which is pluggable and can be applied to existing deep clustering methods. Specifically, we first leverage the pre-trained deep clustering network to compute predictions for all samples, and then design specialized selection strategies to pick out a few high-quality samples as labeled samples for supervised learning. For the unselected samples, the novel unified positive and negative pseudo-label learning is introduced to provide additional supervised signals for semi-supervised fine-tuning. We also propose an adaptive positive-negative threshold learning strategy to further enhance the confidence of generated pseudo-labels. Extensive experiments on six widely-used datasets and one large-scale dataset demonstrate the superiority of our proposed ClusMatch. For example, ClusMatch achieves a significant accuracy improvement of 5.4% over the state-of-the-art method ProPos on an average of these six datasets.

Despite recent promising performances across a range of vision tasks, vision Transformers still have an issue of high computational costs. Recently, vision prompt learning has provided an economical solution to this problem without fine-tuning the whole large-scale model. However, the efficiency and effectiveness of existing models are still far from satisfactory due to the parameter cost of extensive prompt blocks and tricky prompt framework designs. In this paper, we propose a light-weight prompt framework named impLicit vIsion prOmpt tuNing (LION), which is motivated by deep implicit models with stable low memory costs for various complex tasks. In particular, we merely insect two equilibrium implicit layers in two ends of the pre-trained backbone with parameters frozen. Moreover, according to the lottery hypothesis, we further prune the parameters to relieve the computation burden in implicit layers. Various experiments have validated that our LION obtains promising performances on a wide range of datasets. Most importantly, LION reduces up to 11.5 % of training parameter numbers while obtaining higher performance than the state-of-the-art VPT, especially under challenging scenes. Furthermore, we find that our proposed LION has an excellent generalization performance, making it an easy way to boost transfer learning in the future.

Convolutional neural networks benefit from translation equivariance, achieving tremendous success. Equivariant networks further extend this property to other transformation groups. However, most existing methods require dis-cretization or sampling of groups, leading to increased model sizes for larger groups, such as the affine group. In this paper, we build affine equivariant networks based on differential invariants from the viewpoint of symmetric PDEs, without discretizing or sampling the group. To ad-dress the division-by-zero issue arising from fractional dif-ferential invariants of the affine group, we construct a new kind of affine invariants by normalizing polynomial relative differential invariants to replace classical differential invariants. For further flexibility, we design an equivariant layer, which can be directly integrated into convolutional networks of various architectures. Moreover, our frame-work for the affine group is also applicable to its continu-ous subgroups. We implement equivariant networks for the scale group, the rotation-scale group, and the affine group. Numerical experiments demonstrate the outstanding performance of our framework across classification tasks involving transformations of these groups. Remarkably, under the out-of-distribution setting, our model achieves a 3.37% im-provement in accuracy over the main counterpart affConv on the affNIST dataset.

Sequence modeling is a critical task in various domains such as natural language processing, speech recognition, and time series analysis. The existing models still face challenges in capturing long-range dependencies and efficiently modeling sequences. This paper proposes a novel hybrid sequence modeling architecture called MixCon to address these challenges. The MixCon (Mixture of Conba) architecture combines a Transformer layer based on attention mechanism, a Conba layer, and a Mixture of Experts (MoE) module. We apply this idea to the design of the attention mechanism, achieving significant improvements in computational efficiency. Additionally, the MixCon architecture integrates feedback and adaptive control mechanism inspired by control theory, providing a new perspective and approach to sequence modeling. The experimental results demonstrate MixCon’s superior throughput, outperforming Mixtral by 4.5 times and Jamba by 1.5 times when processing lengthy sequences of up to 128K tokens on a single A800 80GB GPU. Moreover, MixCon achieves top-tier scores on academic benchmarks, exemplified by its outstanding performance with a score of 87.9% on HellaSwag and 83.4% on WinoGrande, showcasing its capability to excel in complex sequence modeling tasks.

Neuromorphic computing with spiking neural networks is promising for energy-efficient artificial intelligence (AI) applications. However, different from humans who continually learn different tasks in a lifetime, neural network models suffer from catastrophic forgetting. How could neuronal operations solve this problem is an important question for AI and neuroscience. Many previous studies draw inspiration from observed neuroscience phenomena and propose episodic replay or synaptic metaplasticity, but they are not guaranteed to explicitly preserve knowledge for neuron populations. Other works focus on machine learning methods with more mathematical grounding, e.g., orthogonal projection on high dimensional spaces, but there is no neural correspondence for neuromorphic computing. In this work, we develop a new method with neuronal operations based on lateral connections and Hebbian learning, which can protect knowledge by projecting activity traces of neurons into an orthogonal subspace so that synaptic weight update will not interfere with old tasks. We show that Hebbian and anti-Hebbian learning on recurrent lateral connections can effectively extract the principal subspace of neural activities and enable orthogonal projection. This provides new insights into how neural circuits and Hebbian learning can help continual learning, and also how the concept of orthogonal projection can be realized in neuronal systems. Our method is also flexible to utilize arbitrary training methods based on presynaptic activities/traces. Experiments show that our method consistently solves forgetting for spiking neural networks with nearly zero forgetting under various supervised training methods with different error propagation approaches, and outperforms previous approaches under various settings. Our method can pave a solid path for building continual neuromorphic computing systems. The code is available at https://github.com/pkuxmq/HLOP-SNN.

Spiking neural networks (SNNs) are investigated as biologically inspired models of neural computation, distinguished by their computational capability and energy efficiency due to precise spiking times and sparse spikes with event-driven computation. A significant question is how SNNs can emulate human-like graph-based reasoning of concepts and relations, especially leveraging the temporal domain optimally. This paper reveals that SNNs, when amalgamated with synaptic delay and temporal coding, are proficient in executing (knowledge) graph reasoning. It is elucidated that spiking time can function as an additional dimension to encode relation properties via a neural-generalized path formulation. Empirical results highlight the efficacy of temporal delay in relation processing and showcase exemplary performance in diverse graph reasoning tasks. The spiking model is theoretically estimated to achieve $20\times$ energy savings compared to non-spiking counterparts, deepening insights into the capabilities and potential of biologically inspired SNNs for efficient reasoning. The code is available at https://github.com/pkuxmq/GRSNN.

Foundation Models (FMs) have demonstrated remarkable insights into the relational dynamics of the world, leading to the crucial question: *how do these models acquire an understanding of world hybrid relations?* Traditional statistical learning, particularly for prediction problems, may overlook the rich and inherently structured information from the data, especially regarding the relationships between objects. We introduce a mathematical model that formalizes relational learning as hypergraph recovery to study pre-training of FMs. In our framework, the world is represented as a hypergraph, with data abstracted as random samples from hyperedges. We theoretically examine the feasibility of a Pre-Trained Model (PTM) to recover this hypergraph and analyze the data efficiency in a minimax near-optimal style. By integrating rich graph theories into the realm of PTMs, our mathematical framework offers powerful tools for an in-depth understanding of pre-training from a unique perspective and can be used under various scenarios. As an example, we extend the framework to entity alignment in multimodal learning.

The deep equilibrium model (DEQ) generalizes the conventional feedforward neural network by fixing the same weights for each layer block and extending the number of layers to infinity. This novel model directly finds the fixed points of such a forward process as features for prediction. Despite empirical evidence showcasing its efficacy compared to feedforward neural networks, a theoretical understanding for its separation and bias is still limited. In this paper, we take a step by proposing some separations and studying the bias of DEQ in its expressive power and learning dynamics. The results include: (1) A general separation is proposed, showing the existence of a width-$m$ DEQ that any fully connected neural networks (FNNs) with depth $O(m^{\alpha})$ for $\alpha \in (0,1)$ cannot approximate unless its width is sub-exponential in $m$; (2) DEQ with polynomially bounded size and magnitude can efficiently approximate certain steep functions (which has very large derivatives) in $L^{\infty}$ norm, whereas FNN with bounded depth and exponentially bounded width cannot unless its weights magnitudes are exponentially large; (3) The implicit regularization caused by gradient flow from a diagonal linear DEQ is characterized, with specific examples showing the benefits brought by such regularization. From the overall study, a high-level conjecture from our analysis and empirical validations is that DEQ has potential advantages in learning certain high-frequency components.

Training deep networks on large-scale datasets is computationally challenging. This work explores the problem of “how to accelerate adaptive gradient algorithms in a general manner", and proposes an effective Weight-decay-Integrated Nesterov acceleration (Win) to accelerate adaptive algorithms. Taking AdamW and Adam as examples, per iteration, we construct a dynamical loss that combines the vanilla training loss and a dynamic regularizer inspired by proximal point method, and respectively minimize the first- and second-order Taylor approximations of dynamical loss to update variable. This yields our Win acceleration that uses a conservative step and an aggressive step to update, and linearly combines these two updates for acceleration. Next, we extend Win into Win2 which uses multiple aggressive update steps for faster convergence. Then we apply Win and Win2 to the popular LAMB and SGD optimizers. Our transparent derivation could provide insights for other accelerated methods and their integration into adaptive algorithms. Besides, we theoretically justify the faster convergence of Win- and Win2-accelerated AdamW, Adam and LAMB to their non-accelerated counterparts. Experimental results demonstrates the faster convergence speed and superior performance of our Win- and Win2-accelerated AdamW, Adam, LAMB and SGD over their vanilla counterparts on vision classification and language modeling tasks.

Decentralized optimization over time-varying graphs has been increasingly common in modern machine learning with massive data stored on millions of mobile devices, such as in federated learning. This paper revisits the widely used accelerated gradient tracking and extends it to time-varying graphs. We prove that the practical single loop accelerated gradient tracking needs $O((\frac{\gamma}{1-\sigma_{\gamma}})^2\sqrt{\frac{L}{\epsilon}})$ and $O((\frac{\gamma}{1-\sigma_{\gamma}})^{1.5}\sqrt{\frac{L}{\mu}}\log\frac{1}{\epsilon})$ iterations to reach an $\epsilon$-optimal solution over time-varying graphs when the problems are nonstrongly convex and strongly convex, respectively, where $\gamma$ and $\sigma_{\gamma}$ are two common constants charactering the network connectivity, $L$ and $\mu$ are the smoothness and strong convexity constants, respectively, and one iteration corresponds to one gradient oracle call and one communication round. Our convergence rates improve significantly over the ones of $O(\frac{1}{\epsilon^{5/7}})$ and $O((\frac{L}{\mu})^{5/7}\frac{1}{(1-\sigma)^{1.5}}\log\frac{1}{\epsilon})$, respectively, which were proved in the original literature of accelerated gradient tracking only for static graphs, where $\frac{\gamma}{1-\sigma_{\gamma}}$ equals $\frac{1}{1-\sigma}$ when the network is time-invariant. When combining with a multiple consensus subroutine, the dependence on the network connectivity constants can be further improved to $O(1)$ and $O(\frac{\gamma}{1-\sigma_{\gamma}})$ for the gradient oracle and communication round complexities, respectively. When the network is static, by employing the Chebyshev acceleration, our complexities exactly match the lower bounds without hiding any poly-logarithmic factor for both nonstrongly convex and strongly convex problems.

We consider the non-convex non-concave objective function in two-player zero-sum continuous games. The existence of pure Nash equilibrium requires stringent conditions, posing a major challenge for this problem. To circumvent this difficulty, we examine the problem of identifying a mixed Nash equilibrium, where strategies are randomized and characterized by probability distributions over continuous domains. To this end, we propose PArticle-based Primal-dual ALgorithm (PAPAL) tailored for a weakly entropy-regularized min-max optimization over probability distributions. This algorithm employs the stochastic movements of particles to represent the updates of random strategies for the $\epsilon$-mixed Nash equilibrium. We offer a comprehensive convergence analysis of the proposed algorithm, demonstrating its effectiveness. In contrast to prior research that attempted to update particle importance without movements, PAPAL is the first implementable particle-based algorithm accompanied by non-asymptotic quantitative convergence results, running time, and sample complexity guarantees. Our framework contributes novel insights into the particle-based algorithms for continuous min-max optimization in the general non-convex non-concave setting.

Panoptic Part Segmentation (PPS) unifies panoptic and part segmentation into one task. Previous works utilize separate approaches to handle things, stuff, and part predictions without shared computation and task association. We aim to unify these tasks at the architectural level, designing the first end-to-end unified framework, Panoptic-PartFormer. Moreover, we find the previous metric PartPQ biases to PQ. To handle both issues, we first design a meta-architecture that decouples part features and things/stuff features, respectively. We model things, stuff, and parts as object queries and directly learn to optimize all three forms of prediction as a unified mask prediction and classification problem. We term our model as Panoptic-PartFormer. Second, we propose a new metric Part-Whole Quality (PWQ), better to measure this task from pixel-region and part-whole perspectives. It also decouples the errors for part segmentation and panoptic segmentation. Third, inspired by Mask2Former, based on our meta-architecture, we propose Panoptic-PartFormer++ and design a new part-whole cross-attention scheme to boost part segmentation qualities further. We design a new part-whole interaction method using masked cross attention. Finally, extensive ablation studies and analysis demonstrate the effectiveness of both Panoptic-PartFormer and Panoptic-PartFormer++. Compared with previous Panoptic-PartFormer, our Panoptic-PartFormer++ achieves 2% PartPQ and 3% PWQ improvements on the Cityscapes PPS dataset and 5% PartPQ on the Pascal Context PPS dataset. On both datasets, Panoptic-PartFormer++ achieves new state-of-the-art results. Our models can serve as a strong baseline and aid future research in PPS.

Universal approximation capability, also referred to as universality, is an important property of deep neural networks, endowing them with the potency to accurately represent the underlying target function in learning tasks. In practice, the architecture of deep neural networks largely influences the performance of the models. However, most existing methodologies for designing neural architectures, such as the heuristic manual design or neural architecture search, ignore the universal approximation property, thus losing a potential safeguard about the performance. In this paper, we propose a unified framework to design the architectures of deep neural networks with a universality guarantee based on first-order optimization algorithms, where the forward pass is interpreted as the updates of an optimization algorithm. The (explicit or implicit) network is designed by replacing each gradient term in the algorithm with a learnable module similar to a two-layer network or its derivatives. Specifically, we explore the realm of width-bounded neural networks, a common practical scenario, showcasing their universality. Moreover, adding operations of normalization, downsampling, and upsampling does not hurt the universality. To the best of our knowledge, this is the first work that width-bounded networks with universal approximation guarantee can be designed in a principled way. Our framework can inspire a variety of neural architectures including some renowned structures such as ResNet and DenseNet, as well as novel innovations. The experimental results on image classification problems demonstrate that the newly inspired networks are competitive and surpass the baselines of ResNet, DenseNet, as well as the advanced ConvNeXt and ViT, testifying to the effectiveness of our framework.

In deep learning, different kinds of deep networks typically need different optimizers, which have to be chosen after multiple trials, making the training process inefficient. To relieve this issue and consistently improve the model training speed across deep networks, we propose the ADAptive Nesterov momentum algorithm, Adan for short. Adan first reformulates the vanilla Nesterov acceleration to develop a new Nesterov momentum estimation (NME) method, which avoids the extra overhead of computing gradient at the extrapolation point. Then Adan adopts NME to estimate the gradient's first- and second-order moments in adaptive gradient algorithms for convergence acceleration. Besides, we prove that Adan finds an$\epsilon$-approximate first-order stationary point within$\mathcal {O}(\epsilon ^{-3.5})$stochastic gradient complexity on the non-convex stochastic problems (e.g., deep learning problems), matching the best-known lower bound. Extensive experimental results show that Adan consistently surpasses the corresponding SoTA optimizers on vision, language, and RL tasks and sets new SoTAs for many popular networks and frameworks, e.g., ResNet, ConvNext, ViT, Swin, MAE, DETR, GPT-2, Transformer-XL, and BERT. More surprisingly, Adan can use half of the training cost (epochs) of SoTA optimizers to achieve higher or comparable performance on ViT, GPT-2, MAE,etc, and also shows great tolerance to a large range of minibatch size, e.g., from 1 k to 32 k.

AdamW modifies Adam by adding a decoupled weight decay to decay network weights per training iteration. For adaptive algorithms, this decoupled weight decay does not affect specific optimization steps, and differs from the widely used$\ell _{2}$-regularizer which changes optimization steps via changing the first- and second-order gradient moments. Despite its great practical success, for AdamW, its convergence behavior and generalization improvement over Adam and$\ell _{2}$-regularized Adam ($\ell _{2}$-Adam) remain absent yet. To solve this issue, we prove the convergence of AdamW and justify its generalization advantages over Adam and$\ell _{2}$-Adam. Specifically, AdamW provably converges but minimizes a dynamically regularized loss that combines vanilla loss and a dynamical regularization induced by decoupled weight decay, thus yielding different behaviors with Adam and$\ell _{2}$-Adam. Moreover, on both general nonconvex problems and PŁ-conditioned problems, we establish stochastic gradient complexity of AdamW to find a stationary point. Such complexity is also applicable to Adam and$\ell _{2}$-Adam, and improves their previously known complexity, especially for over-parametrized networks. Besides, we prove that AdamW enjoys smaller generalization errors than Adam and$\ell _{2}$-Adam from the Bayesian posterior aspect. This result, for the first time, explicitly reveals the benefits of decoupled weight decay in AdamW. Experimental results validate our theory.

In this article, a curious phenomenon in the tensor recovery algorithm is considered: can the same recovered results be obtained when the observation tensors in the algorithm are transposed in different ways? If not, it is reasonable to imagine that some information within the data will be lost for the case of observation tensors under certain transpose operators. To solve this problem, a new tensor rank called weighted tensor average rank (WTAR) is proposed to learn the relationship between different resulting tensors by performing a series of transpose operators on an observation tensor. WTAR is applied to three-order tensor robust principal component analysis (TRPCA) to investigate its effectiveness. Meanwhile, to balance the effectiveness and solvability of the resulting model, a generalized model that involves the convex surrogate and a series of nonconvex surrogates are studied, and the corresponding worst case error bounds of the recovered tensor is given. Besides, a generalized tensor singular value thresholding (GTSVT) method and a generalized optimization algorithm based on GTSVT are proposed to solve the generalized model effectively. The experimental results indicate that the proposed method is effective.

A deep equilibrium model (DEQ) is implicitly defined through an equilibrium point of an infinite-depth weight-tied model with an input-injection. Instead of infinite computations, it solves an equilibrium point directly with root-finding and computes gradients with implicit differentiation. In this paper, the training dynamics of over-parameterized DEQs are investigated, and we propose a novel probabilistic framework to overcome the challenge arising from the weight-sharing and the infinite depth. By supposing a condition on the initial equilibrium point, we prove that the gradient descent converges to a globally optimal solution at a linear convergence rate for the quadratic loss function. We further perform a fine-grained non-asymptotic analysis about random DEQs and the corresponding weight-untied models, and show that the required initial condition is satisfied via mild over-parameterization. Moreover, we show that the unique equilibrium point always exists during the training.

Polyak-Łojasiewicz (PL) (Polyak, 1963) condition is a weaker condition than the strong convexity but suffices to ensure a global convergence for the Gradient Descent algorithm. In this paper, we study the lower bound of algorithms using first-order oracles to find an approximate optimal solution. We show that any first-order algorithm requires at least ${\Omega}\left(\frac{L}{\mu}\log\frac{1}{\epsilon}\right)$ gradient costs to με find an $\epsilon$-approximate optimal solution for a general $L$-smooth function that has an $\mu$-PL constant. This result demonstrates the optimality of the Gradient Descent algorithm to minimize smooth PL functions in the sense that there exists a “hard” PL function such that no first-order algorithm can be faster than Gradient Descent when ignoring a numerical constant. In contrast, it is well-known that the momentum technique, e.g. Nesterov (2003, chap. 2), can provably accelerate Gradient Descent to ${O}\left(\sqrt{\frac{L}{\hat{\mu}}}\log\frac{1}{\epsilon}\right)$ gradient costs for functions that are $L$-smooth and $\hat{\mu}$-strongly convex. Therefore, our result distinguishes the hardness of minimizing a smooth PL function and a smooth strongly convex function as the complexity of the former cannot be improved by any polynomial order in general.

Zeroth-order optimization is a fundamental research topic that has been a focus of various learning tasks, such as black-box adversarial attacks, bandits, and reinforcement learning. However, in theory, most complexity results assert a linear dependency on the dimension of optimization variable, which implies paralyzations of zeroth-order algorithms for high-dimensional problems and cannot explain their effectiveness in practice. In this paper, we present a novel zeroth-order optimization theory characterized by complexities that exhibit weak dependencies on dimensionality. The key contribution lies in the introduction of a new factor, denoted as $\effdim_{\alpha} = \sup_{\x\in \mathbb{R}^d} \sum_{i=1}^d \sigma_i^\alpha(\nabla^2 f(\x))$ ($\alpha>0$, $\sigma_i(\cdot)$ is the $i$-th singular value in non-increasing order), which effectively functions as a measure of dimensionality. The algorithms we propose demonstrate significantly reduced complexities when measured in terms of the factor $\effdim_{\alpha}$. Specifically, we first study a well-known zeroth-order algorithm from Nesterov and Spokoiny (2017) on quadratic objectives and show a complexity of $ \mathcal{O}\left(\frac{\effdim_1 }{\sigma_d}\log(1/\epsilon)\right) $ for the strongly convex setting. For linear regression, such a complexity is dimension-free and outperforms the traditional result by a factor of $d$ under common conditions. Furthermore, we introduce novel algorithms that leverages the Heavy-ball mechanism to enhance the optimization process. By incorporating this acceleration scheme, our proposed algorithm exhibits a complexity of $ \mathcal{O}\left(\frac{\effdim_{1/2} }{\sqrt{\sigma_d}}\cdot\log{\frac{L}{\mu}}\cdot\log(1/\epsilon)\right) $. For linear regression, under some mild conditions, it is faster than state-of-the-art algorithms by $\sqrt{d}$. We further expand the scope of the method to encompass generic smooth optimization problems, while incorporating an additional Hessian-smooth condition. By considering this extended framework, our approach becomes applicable to a broader range of optimization scenarios. The resultant algorithms demonstrate remarkable complexities, with dimension-independent dominant terms that surpass existing algorithms by an order in $d$ under appropriate conditions. Our analysis lays the foundation for investigating zeroth-order optimization methods for smooth functions within high-dimensional settings.

Spiking Neural Networks (SNNs) are promising energy-efficient models for neuromorphic computing. For training the non-differentiable SNN methods, the backpropagation through time (BPTT) with surrogate gradients (SG) method has achieved high performance. However, this method suffers from considerable memory cost and training time during training. In this paper, we propose the Spatial Learning Through Time (SLTT) method that can achieve high performance while greatly improving training efficiency compared with BPTT. First, we show that the backpropagation of SNNs through the temporal domain contributes just a little to the final calculated gradients. Thus, we propose to ignore the unimportant routes in the computational graph during backpropagation. The proposed method reduces the number of scalar multiplications and achieves a small memory occupation that is independent of the total time steps. Furthermore, we propose a variant of SLTT, called SLTT-K, that allows backpropagation only at K time steps, then the required number of scalar multiplications is further reduced and is independent of the total time steps. Experiments on both static and neuromorphic datasets demonstrate superior training efficiency and performance of our SLTT. In particular, our method achieves state-of-the-art accuracy on ImageNet, while the memory cost and training time are reduced by more than 70% and 50%, respectively, compared with BPTT. Our code is available at https://github.com/qymeng94/SLTT.

In recent years, graph Transformers (GTs) have been demonstrated as a robust architecture for a wide range of graph learning tasks. However, the quadratic complexity of GTs limits their scalability on large-scale data, in comparison to Graph Neural Networks (GNNs). In this work, we propose the Kernel Decomposition Linear Graph Transformer (KDLGT), an accelerating framework for building scalable and powerful GTs. KDLGT employs the kernel decomposition approach to rearrange the order of matrix multiplication, thereby reducing complexity to linear. Additionally, it categorizes GTs into three distinct types and provides tailored accelerating methods for each category to encompass all types of GTs. Furthermore, we provide a theoretical analysis of the performance gap between KDLGT and self-attention to ensure its effectiveness. Under this framework, we select two representative GTs to design our models. Experiments on both real-world and synthetic datasets indicate that KDLGT not only achieves state-of-the-art performance on various datasets but also reaches an acceleration ratio of approximately 10 on graphs of certain sizes.

Recent empirical studies observe that even without normalization, a deep residual network can be trained reliably. We call such a structure as normalization-free Residual Networks (N-F ResNets), which add a learnable parameter$\alpha$to control the scale of the residual block instead of normalization. However, the theoretical understanding on N-F ResNets is still limited despite their empirical success. In this paper, we provide the first theoretical understanding of N-F ResNets from two perspectives. Firstly, we prove that the gradient descent (GD) algorithm can find the global minimum of the training loss at a linear rate for over-parameterized N-F ResNets. Secondly, we prove that N-F ResNets can avoid the gradient exploding or vanishing problem, by initializing the key parameter$\alpha$to be a small constant. Notably, we demonstrate that the gradients of N-F ResNets are more stable than those of ResNets with Kaiming initialization. Moreover, empirical experiments on benchmark datasets verify our theoretical results.

In this paper, we propose a novel extra-gradient difference acceleration algorithm for solving constrained nonconvex-nonconcave (NC-NC) minimax problems. In particular, we design a new extra-gradient difference step to obtain an important quasi-cocoercivity property, which plays a key role to significantly improve the convergence rate in the constrained NC-NC setting without additional structural assumption. Then momentum acceleration is also introduced into our dual accelerating update step. Moreover, we prove that, to find an $\epsilon$-stationary point of the function $f$, our algorithm attains the complexity $\mathcal{O}(\epsilon^{-2})$ in the constrained NC-NC setting, while the best-known complexity bound is $\widetilde{\mathcal{O}}(\epsilon^{-4})$, where $\widetilde{\mathcal{O}}(\cdot)$ hides logarithmic factors compared to $\mathcal{O}(\cdot)$. As the special cases of the constrained NC-NC setting, our algorithm can also obtain the same complexity $\mathcal{O}(\epsilon^{-2})$ for both the nonconvex-concave (NC-C) and convex-nonconcave (C-NC) cases, while the best-known complexity bounds are $\widetilde{\mathcal{O}}(\epsilon^{-2.5})$ for the NC-C case and $\widetilde{\mathcal{O}}(\epsilon^{-4})$ for the C-NC case. For fair comparison with existing algorithms, we also analyze the complexity bound to find $\epsilon$-stationary point of the primal function $\phi$ for the constrained NC-C problem, which shows that our algorithm can improve the complexity bound from $\widetilde{\mathcal{O}}(\epsilon^{-3})$ to $\mathcal{O}(\epsilon^{-2})$. To the best of our knowledge, this is the first time that the proposed algorithm improves the best-known complexity bounds from $\mathcal{O}(\epsilon^{-4})$ and $\widetilde{\mathcal{O}}(\epsilon^{-3})$ to $\mathcal{O}(\epsilon^{-2})$ in both the NC-NC and NC-C settings.

Despite the connection established by optimization-induced deep equilibrium models (OptEqs) between their output and the underlying hidden optimization problems, the performance of it along with its related works is still not good enough especially when compared to deep networks. One key factor responsible for this performance limitation is the use of linear kernels to extract features in these models. To address this issue, we propose a novel approach by replacing its linear kernel with a new function that can readily capture nonlinear feature dependencies in the input data. Drawing inspiration from classical machine learning algorithms, we introduce Gaussian kernels as the alternative function and then propose our new equilibrium model, which we refer to as GEQ. By leveraging Gaussian kernels, GEQ can effectively extract the nonlinear information embedded within the input features, surpassing the performance of the original OptEqs. Moreover, GEQ can be perceived as a weight-tied neural network with infinite width and depth. GEQ also enjoys better theoretical properties and improved overall performance. Additionally, our GEQ exhibits enhanced stability when confronted with various samples. We further substantiate the effectiveness and stability of GEQ through a series of comprehensive experiments.

Pre-training has achieved remarkable success when transferred to downstream tasks. In machine learning, we care about not only the good performance of a model but also its behavior under reasonable shifts of condition. The same philosophy holds when pre-training a foundation model. However, the foundation model may not uniformly behave well for a series of related downstream tasks. This happens, for example, when conducting mask recovery regression where the recovery ability or the training instances diverge like pattern features are extracted dominantly on pre-training, but semantic features are also required on a downstream task. This paper considers pre-training a model that guarantees a uniformly good performance over the downstream tasks. We call this goal as *downstream-task robustness*. Our method first separates the upstream task into several representative ones and applies a simple minimax loss for pre-training. We then design an efficient algorithm to solve the minimax loss and prove its convergence in the convex setting. In the experiments, we show both on large-scale natural language processing and computer vision datasets our method increases the metrics on worse-case downstream tasks. Additionally, some theoretical explanations for why our loss is beneficial are provided. Specifically, we show fewer samples are inherently required for the most challenging downstream task in some cases.

Adversarial Training (AT) has become arguably the state-of-the-art algorithm for extracting robust features. However, researchers recently notice that AT suffers from severe robust overfitting problems, particularly after learning rate (LR) decay. In this paper, we explain this phenomenon by viewing adversarial training as a dynamic minimax game between the model trainer and the attacker. Specifically, we analyze how LR decay breaks the balance between the minimax game by empowering the trainer with a stronger memorization ability, and show such imbalance induces robust overfitting as a result of memorizing non-robust features. We validate this understanding with extensive experiments, and provide a holistic view of robust overfitting from the dynamics of both the two game players. This understanding further inspires us to alleviate robust overfitting by rebalancing the two players by either regularizing the trainer's capacity or improving the attack strength. Experiments show that the proposed ReBalanced Adversarial Training (ReBAT) can attain good robustness and does not suffer from robust overfitting even after very long training. Code is available at https://github.com/PKU-ML/ReBAT.

This paper studies accelerated gradient methods for nonconvex optimization with Lipschitz continuous gradient and Hessian. We propose two simple accelerated gradient methods, restarted accelerated gradient descent (AGD) and restarted heavy ball (HB) method, and establish that our methods achieve an $\epsilon$-approximate first-order stationary point within $O(\epsilon^{-7/4})$ number of gradient evaluations by elementary proofs. Theoretically, our complexity does not hide any polylogarithmic factors, and thus it improves over the best known one by the $O(\log\frac{1}{\epsilon})$ factor. Our algorithms are simple in the sense that they only consist of Nesterov's classical AGD or Polyak's HB iterations, as well as a restart mechanism. They do not invoke negative curvature exploitation or minimization of regularized surrogate functions as the subroutines. In contrast with existing analysis, our elementary proofs use less advanced techniques and do not invoke the analysis of strongly convex AGD or HB.

To reveal the mystery behind deep neural networks (DNNs), optimization may offer a good perspective. There are already some clues showing the strong connection between DNNs and optimization problems, e.g., under a mild condition, DNN's activation function is indeed a proximal operator. In this paper, we are committed to providing a unified optimization induced interpretability for a special class of networks-equilibrium models, i.e., neural networks defined by fixed point equations, which have become increasingly attractive recently. To this end, we first decompose DNNs into a new class of unit layer that is the proximal operator of an implicit convex function while keeping its output unchanged. Then, the equilibrium model of the unit layer can be derived, we name it Optimization Induced Equilibrium Networks (OptEq). The equilibrium point of OptEq can be theoretically connected to the solution of a convex optimization problem with explicit objectives. Based on this, we can flexibly introduce prior properties to the equilibrium points: 1) modifying the underlying convex problems explicitly so as to change the architectures of OptEq; and 2) merging the information into the fixed point iteration, which guarantees to choose the desired equilibrium point when the fixed point set is non-singleton. We show that OptEq outperforms previous implicit models even with fewer parameters.

In this paper, we present a general optimization framework that leverages structured sparsity to achieve superior recovery results. The traditional method for solving the structured sparse objectives based on$\ell _{2,0}$-norm is to use the$\ell _{2,1}$-norm as a convex surrogate. However, such an approximation often yields a large performance gap. To tackle this issue, we first provide a framework that allows for a wide range of surrogate functions (including non-convex surrogates), which exhibits better performance in harnessing structured sparsity. Moreover, we develop a fixed point algorithm that solves a key underlying non-convex structured sparse recovery optimization problem to global optimality with a guaranteed super-linear convergence rate. Building on this, we consider three specific applications, i.e., outlier pursuit, supervised feature selection, and structured dictionary learning, which can benefit from the proposed structured sparsity optimization framework. In each application, how the optimization problem can be formulated and thus be relaxed under a generic surrogate function is explained in detail. We conduct extensive experiments on both synthetic and real-world data and demonstrate the effectiveness and efficiency of the proposed framework.

Recent efforts on learning-based image denoising approaches use unrolled architectures with a fixed number of repeatedly stacked blocks. However, due to difficulties in training networks corresponding to deeper layers, simply stacking blocks may cause performance degradation, and the number of unrolled blocks needs to be manually tuned to find an appropriate value. To circumvent these problems, this paper describes an alternative approach with implicit models. To our best knowledge, our approach is the first attempt to model iterative image denoising through an implicit scheme. The model employs implicit differentiation to calculate gradients in the backward pass, thus avoiding the training difficulties of explicit models and elaborate selection of the iteration number. Our model is parameter-efficient and has only one implicit layer, which is a fixed-point equation that casts the desired noise feature as its solution. By simulating infinite iterations of the model, the final denoising result is given by the equilibrium that is achieved through accelerated black-box solvers. The implicit layer not only captures the non-local self-similarity prior for image denoising, but also facilitates training stability and thereby boosts the denoising performance. Extensive experiments show that our model leads to better performances than state-of-the-art explicit denoisers with enhanced qualitative and quantitative results.

On large-scale graphs, many graph neural networks are problematic in capturing long-range dependencies due to the oversmoothing problem. Recently, Graph Equilibrium Models (GEQs) arise as a promising solution to this issue. Their output is the equilibrium of a fixed-point equation, which can be seen as the result of iterating a GNN layer for infinite times, so that they inherently have global receptive fields. However, to find the equilibrium, GEQs require running costly full-batch root-finding algorithms from scratch during each model update, which leads to severe efficiency and scalability issues that prevent them from scaling to large graphs. To address these limitations, we propose VEQ, an efficient learning method to scale GEQs to large graphs. Instead of initializing the equilibrium from scratch in full-batch training, VEQ uses the latest equilibrium of in-batch nodes and their 1-hop neighbors (dubbed Virtual Equilibrium) to accelerate and calibrate the root-finding process in mini-batch training. With virtual equilibrium as an informative prior, VEQ is able to reach the equilibrium in fewer steps while still capturing global dependencies. Theoretically, we provide convergence analysis for the forward and backward pass of VEQ. Empirically, VEQ significantly outperforms existing GEQs by a large margin (more than 1.5%) on all benchmark datasets, with much less training time and memory. Also, VEQ achieves competitive and even superior performance to many highly engineered explicit GNNs on large-scale benchmark datasets like ogbn-arxiv and ogbn-products. VEQ shows that after we resolve the efficiency and scalability issues, GEQs are indeed favorable on large graphs due to their advantage of capturing long-range dependencies.

Epilepsy is a chronic neurological disease that affects many people in the world. Automatic epileptic seizure detection based on electroencephalogram (EEG) signals is of great significance and has been widely studied. The current deep learning epilepsy detection algorithms are often designed to be relatively simple and seldom consider the characteristics of EEG signals. In this paper, we propose a promising epilepsy detection model based on convolutional transformer networks. We demonstrate that integrating convolution and transformer modules can achieve higher detection performance. Our convolutional transformer model is composed of two branches: one extracts time-domain features from multiple inputs of channel-exchanged EEG signals, and the other handle frequency-domain representations. Experiments on two EEG datasets show that our model offers state-of-the-art performance. Particularly on the CHB-MIT dataset, our model achieves 96.02% in average sensitivity and 97.94% in average specificity, outperforming other existing methods with clear margins.

Spiking Neural Network (SNN) is a promising energy-efficient AI model when implemented on neuromorphic hardware. However, it is a challenge to efficiently train SNNs due to their non-differentiability. Most existing methods either suffer from high latency (i.e., long simulation time steps), or cannot achieve as high performance as Artificial Neural Networks (ANNs). In this paper, we propose the Differentiation on Spike Representation (DSR) method, which could achieve high performance that is competitive to ANNs yet with low latency. First, we encode the spike trains into spike representation using (weighted) firing rate coding. Based on the spike representation, we systematically derive that the spiking dynamics with common neural models can be represented as some sub-differentiable mapping. With this viewpoint, our proposed DSR method trains SNNs through gradients of the mapping and avoids the common non-differentiability problem in SNN training. Then we analyze the error when representing the specific mapping with the forward computation of the SNN. To reduce such error, we propose to train the spike threshold in each layer, and to introduce a new hyperparameter for the neural models. With these components, the DSR method can achieve state-of-the-art SNN performance with low latency on both static and neuromorphic datasets, including CIFAR-10, CIFAR-100, ImageNet, and DVS-CIFAR10.

Recently, some accelerated stochastic variance reduction algorithms such as Katyusha and ASVRG-ADMM achieve faster convergence than non-accelerated methods such as SVRG and SVRG-ADMM. However, there are still some gaps between the oracle complexities and their lower bounds. To fill in these gaps, this paper proposes a novel Directly Accelerated stochastic Variance reductIon (DAVIS) algorithm with two Snapshots for non-strongly convex (non-SC) unconstrained problems. Our theoretical results show that DAVIS achieves the optimal convergence rate O(1/(nS^2)) and optimal gradient complexity O(n+\sqrt{nL/\epsilon}), which is identical to its lower bound. To the best of our knowledge, this is the first directly accelerated algorithm that attains the optimal lower bound and improves the convergence rate from O(1/S^2) to O(1/(nS^2)). Moreover, we extend DAVIS and theoretical results to non-SC problems with a structured regularizer, and prove that the proposed algorithm with double-snapshots also attains the optimal convergence rate O(1/(nS)) and optimal oracle complexity O(n+L/\epsilon) for such problems, and it is at least a factor n/S faster than existing accelerated stochastic algorithms, where n\gg S in general.

This paper studies the accelerated gradient descent for general nonconvex problems under the gradient Lipschitz and Hessian Lipschitz assumptions. We establish that a simple restarted accelerated gradient descent (AGD) finds an $\epsilon$-approximate first-order stationary point in $O(\epsilon^{-7/4})$ gradient computations with simple proofs. Our complexity does not hide any polylogarithmic factors, and thus it improves over the state-of-the-art one by the $O(\log\frac{1}{\epsilon})$ factor. Our simple algorithm only consists of Nesterov’s classical AGD and a restart mechanism, and it does not need the negative curvature exploitation or the optimization of regularized surrogate functions. Technically, our simple proof does not invoke the analysis for the strongly convex AGD, which is crucial to remove the $O(\log\frac{1}{\epsilon})$ factor.

Due to the over-smoothing issue, most existing graph neural networks can only capture limited dependencies with their inherently finite aggregation layers. To overcome this limitation, we propose a new kind of graph convolution, called Graph Implicit Nonlinear Diffusion (GIND), which implicitly has access to infinite hops of neighbors while adaptively aggregating features with nonlinear diffusion to prevent over-smoothing. Notably, we show that the learned representation can be formalized as the minimizer of an explicit convex optimization objective. With this property, we can theoretically characterize the equilibrium of our GIND from an optimization perspective. More interestingly, we can induce new structural variants by modifying the corresponding optimization objective. To be specific, we can embed prior properties to the equilibrium, as well as introducing skip connections to promote training stability. Extensive experiments show that GIND is good at capturing long-range dependencies, and performs well on both homophilic and heterophilic graphs with nonlinear diffusion. Moreover, we show that the optimization-induced variants of our models can boost the performance and improve training stability and efficiency as well. As a result, our GIND obtains significant improvements on both node-level and graph-level tasks.

Recently, certifiable robust training methods via bound propagation have been proposed for training neural networks with certifiable robustness guarantees. However, no neural architectures with regular convolution and linear layers perform better in the certifiable training than the plain CNNs, since the output bounds for the deep explicit models increase quickly as their depth increases. And such a phenomenon significantly hinders certifiable training. Meanwhile, the Deep Equilibrium Model (DEQ) is more representative and robust due to their equivalent infinite depth and controllable global Lipschitz. But no work has been proposed to explore whether DEQ can show advantages in certified training. In this work, we aim to tackle the problem of DEQ’s certified training. To obtain the output bound based on the bound propagation scheme in the implicit model, we first involve the adjoint DEQ for bound approximation. Furthermore, we also use the weight orthogonalization method and other tricks specified for DEQ to stabilize the certifiable training. With our approach, we can obtain the certifiable DEQ called CerDEQ. Our CerDEQ can achieve state-of-the-art performance compared with models using regular convolution and linear layers on $\ell_\infty$ tasks with $\epsilon=8/255$: $64.72%$ certified error for CIFAR-$10$ and $94.45%$ certified error for Tiny ImageNet.

Recently, linear GCNs have shown competitive performance against non-linear ones with less computation cost, and the key lies in their propagation layers. Spectral analysis has been widely adopted in designing and analyzing existing graph propagations. Nevertheless, we notice that existing spectral analysis fails to explain why existing graph propagations with the same global tendency, such as low-pass or high-pass, still yield very different results. Motivated by this situation, we develop a new framework for spectral analysis in this paper called concentration analysis. In particular, we propose three attributes: concentration centre, maximum response, and bandwidth for our analysis. Through a dissection of the limitations of existing graph propagations via the above analysis, we propose a new kind of propagation layer, Graph Gaussian Convolution Networks (G^2CN), in which the three properties are decoupled and the whole structure becomes more flexible and applicable to different kinds of graphs. Extensive experiments show that we can obtain state-of-the-art performance on heterophily and homophily datasets with our proposed G^2CN.

Steerable models can provide very general and flexible equivariance by formulating equivariance requirements in the language of representation theory and feature fields, which has been recognized to be effective for many vision tasks. However, deriving steerable models for 3D rotations is much more difficult than that in the 2D case, due to more complicated mathematics of 3D rotations. In this work, we employ partial differential operators (PDOs) to model 3D filters, and derive general steerable 3D CNNs, which are called PDO-s3DCNNs. We prove that the equivariant filters are subject to linear constraints, which can be solved efficiently under various conditions. As far as we know, PDO-s3DCNNs are the most general steerable CNNs for 3D rotations, in the sense that they cover all common subgroups of SO(3) and their representations, while existing methods can only be applied to specific groups and representations. Extensive experiments show that our models can preserve equivariance well in the discrete domain, and outperform previous works on SHREC’17 retrieval and ISBI 2012 segmentation tasks with a low network complexity.

Spiking neural networks (SNNs) are promising brain-inspired energy-efficient models. Recent progress in training methods has enabled successful deep SNNs on large-scale tasks with low latency. Particularly, backpropagation through time (BPTT) with surrogate gradients (SG) is popularly used to enable models to achieve high performance in a very small number of time steps. However, it is at the cost of large memory consumption for training, lack of theoretical clarity for optimization, and inconsistency with the online property of biological learning rules and rules on neuromorphic hardware. Other works connect the spike representations of SNNs with equivalent artificial neural network formulation and train SNNs by gradients from equivalent mappings to ensure descent directions. But they fail to achieve low latency and are also not online. In this work, we propose online training through time (OTTT) for SNNs, which is derived from BPTT to enable forward-in-time learning by tracking presynaptic activities and leveraging instantaneous loss and gradients. Meanwhile, we theoretically analyze and prove that the gradients of OTTT can provide a similar descent direction for optimization as gradients from equivalent mapping between spike representations under both feedforward and recurrent conditions. OTTT only requires constant training memory costs agnostic to time steps, avoiding the significant memory costs of BPTT for GPU training. Furthermore, the update rule of OTTT is in the form of three-factor Hebbian learning, which could pave a path for online on-chip learning. With OTTT, it is the first time that the two mainstream supervised SNN training methods, BPTT with SG and spike representation-based training, are connected, and meanwhile it is in a biologically plausible form. Experiments on CIFAR-10, CIFAR-100, ImageNet, and CIFAR10-DVS demonstrate the superior performance of our method on large-scale static and neuromorphic datasets in a small number of time steps. Our code is available at https://github.com/pkuxmq/OTTT-SNN.

Modern deep neural networks for classification usually jointly learn a backbone for representation and a linear classifier to output the logit of each class. A recent study has shown a phenomenon called neural collapse that the within-class means of features and the classifier vectors converge to the vertices of a simplex equiangular tight frame (ETF) at the terminal phase of training on a balanced dataset. Since the ETF geometric structure maximally separates the pair-wise angles of all classes in the classifier, it is natural to raise the question, why do we spend an effort to learn a classifier when we know its optimal geometric structure? In this paper, we study the potential of learning a neural network for classification with the classifier randomly initialized as an ETF and fixed during training. Our analytical work based on the layer-peeled model indicates that the feature learning with a fixed ETF classifier naturally leads to the neural collapse state even when the dataset is imbalanced among classes. We further show that in this case the cross entropy (CE) loss is not necessary and can be replaced by a simple squared loss that shares the same global optimality but enjoys a better convergence property. Our experimental results show that our method is able to bring significant improvements with faster convergence on multiple imbalanced datasets.

Most knowledge graphs (KGs) are incomplete, which motivates one important research topic on automatically complementing knowledge graphs. However, evaluation of knowledge graph completion (KGC) models often ignores the incompleteness---facts in the test set are ranked against all unknown triplets which may contain a large number of missing facts not included in the KG yet. Treating all unknown triplets as false is called the closed-world assumption. This closed-world assumption might negatively affect the fairness and consistency of the evaluation metrics. In this paper, we study KGC evaluation under a more realistic setting, namely the open-world assumption, where unknown triplets are considered to include many missing facts not included in the training or test sets. For the currently most used metrics such as mean reciprocal rank (MRR) and Hits@K, we point out that their behavior may be unexpected under the open-world assumption. Specifically, with not many missing facts, their numbers show a logarithmic trend with respect to the true strength of the model, and thus, the metric increase could be insignificant in terms of reflecting the true model improvement. Further, considering the variance, we show that the degradation in the reported numbers may result in incorrect comparisons between different models, where stronger models may have lower metric numbers. We validate the phenomenon both theoretically and experimentally. Finally, we suggest possible causes and solutions for this problem. Our code and data are available at https://github.com/GraphPKU/Open-World-KG .

Automated machine learning has been widely explored to reduce human efforts in designing neural architectures and looking for proper hyperparameters. In the domain of neural initialization, however, similar automated techniques have rarely been studied. Most existing initialization methods are handcrafted and highly dependent on specific architectures. In this paper, we propose a differentiable quantity, named GradCoisne, with theoretical insights to evaluate the initial state of a neural network. Specifically, GradCosine is the cosine similarity of sample-wise gradients with respect to the initialized parameters. By analyzing the sample-wise optimization landscape, we show that both the training and test performance of a network can be improved by maximizing GradCosine under gradient norm constraint. Based on this observation, we further propose the neural initialization optimization (NIO) algorithm. Generalized from the sample-wise analysis into the real batch setting, NIO is able to automatically look for a better initialization with negligible cost compared with the training time. With NIO, we improve the classification performance of a variety of neural architectures on CIFAR10, CIFAR-100, and ImageNet. Moreover, we find that our method can even help to train large vision Transformer architecture without warmup.

We study stochastic decentralized optimization for the problem of training machine learning models with large-scale distributed data. We extend the widely used EXTRA and DIGing methods with variance reduction (VR), and propose two methods: VR-EXTRA and VR-DIGing. The proposed VR-EXTRA requires the time of $O((\kappa_s+n)\log\frac{1}{\epsilon})$ stochastic gradient evaluations and $O((\kappa_b+\kappa_c)\log\frac{1}{\epsilon})$ communication rounds to reach precision $\epsilon$, which are the best complexities among the non-accelerated gradient-type methods, where $\kappa_s$ and $\kappa_b$ are the stochastic condition number and batch condition number for strongly convex and smooth problems, respectively, $\kappa_c$ is the condition number of the communication network, and $n$ is the sample size on each distributed node. The proposed VR-DIGing has a little higher communication cost of $O((\kappa_b+\kappa_c^2)\log\frac{1}{\epsilon})$. Our stochastic gradient computation complexities are the same as the ones of single-machine VR methods, such as SAG, SAGA, and SVRG, and our communication complexities keep the same as those of EXTRA and DIGing, respectively. To further speed up the convergence, we also propose the accelerated VR-EXTRA and VR-DIGing with both the optimal $O((\sqrt{n\kappa_s}+n)\log\frac{1}{\epsilon})$ stochastic gradient computation complexity and $O(\sqrt{\kappa_b\kappa_c}\log\frac{1}{\epsilon})$ communication complexity. Our stochastic gradient computation complexity is also the same as the ones of single-machine accelerated VR methods, such as Katyusha, and our communication complexity keeps the same as those of accelerated full batch decentralized methods, such as MSDA. To the best of our knowledge, our accelerated methods are the first to achieve both the optimal stochastic gradient computation complexity and communication complexity in the class of gradient-type methods.

In this paper, we propose an ensemble learning algorithm called under-bagging $k$-nearest neighbors (under-bagging $k$-NN) for imbalanced classification problems. On the theoretical side, by developing a new learning theory analysis, we show that with properly chosen parameters, i.e., the number of nearest neighbors $k$, the expected sub-sample size $s$, and the bagging rounds $B$, optimal convergence rates for under-bagging $k$-NN can be achieved under mild assumptions w.r.t. the arithmetic mean (AM) of recalls. Moreover, we show that with a relatively small $B$, the expected sub-sample size $s$ can be much smaller than the number of training data $n$ at each bagging round, and the number of nearest neighbors $k$ can be reduced simultaneously, especially when the data are highly imbalanced, which leads to substantially lower time complexity and roughly the same space complexity. On the practical side, we conduct numerical experiments to verify the theoretical results on the benefits of the under-bagging technique by the promising AM performance and efficiency of our proposed algorithm.

We present the lifted proximal operator machine (LPOM) to train fully-connected feed-forward neural networks. LPOM represents the activation function as an equivalent proximal operator and adds the proximal operators to the objective function of a network as penalties. LPOM is block multi-convex in all layer-wise weights and activations. This allows us to develop a new block coordinate descent (BCD) method with convergence guarantee to solve it. Due to the novel formulation and solving method, LPOM only uses the activation function itself and does not require any gradient steps. Thus it avoids the gradient vanishing or exploding issues, which are often blamed in gradient-based methods. Also, it can handle various non-decreasing Lipschitz continuous activation functions. Additionally, LPOM is almost as memory-efficient as stochastic gradient descent and its parameter tuning is relatively easy. We further implement and analyze the parallel solution of LPOM. We first propose a general asynchronous-parallel BCD method with convergence guarantee. Then we use it to solve LPOM, resulting in asynchronous-parallel LPOM. For faster speed, we develop the synchronous-parallel LPOM. We validate the advantages of LPOM on various network architectures and datasets. We also apply synchronous-parallel LPOM to autoencoder training and demonstrate its fast convergence and superior performance.

Bi-Level Optimization (BLO) is originated from the area of economic game theory and then introduced into the optimization community. BLO is able to handle problems with a hierarchical structure, involving two levels of optimization tasks, where one task is nested inside the other. In machine learning and computer vision fields, despite the different motivations and mechanisms, a lot of complex problems, such as hyper-parameter optimization, multi-task and meta learning, neural architecture search, adversarial learning and deep reinforcement learning, actually all contain a series of closely related subproblms. In this paper, we first uniformly express these complex learning and vision problems from the perspective of BLO. Then we construct a best-response-based single-level reformulation and establish a unified algorithmic framework to understand and formulate mainstream gradient-based BLO methodologies, covering aspects ranging from fundamental automatic differentiation schemes to various accelerations, simplifications, extensions and their convergence and complexity properties. Last but not least, we discuss the potentials of our unified BLO framework for designing new algorithms and point out some promising directions for future research. A list of important papers discussed in this survey, corresponding codes, and additional resources on BLOs are publicly available at: https://github.com/vis-opt-group/BLO.

Sparsity-constrained optimization problems are common in machine learning, such as sparse coding, low-rank minimization and compressive sensing. However, most of previous studies focused on constructing various hand-crafted sparse regularizers, while little work was devoted to learning adaptive sparse regularizers from given input data for specific tasks. In this paper, we propose a deep sparse regularizer learning model that learns data-driven sparse regularizers adaptively. Via the proximal gradient algorithm, we find that the sparse regularizer learning is equivalent to learning a parameterized activation function. This encourages us to learn sparse regularizers in the deep learning framework. Therefore, we build a neural network composed of multiple blocks, each being differentiable and reusable. All blocks contain learnable piecewise linear activation functions which correspond to the sparse regularizer to be learned. Furthermore, the proposed model is trained with back propagation, and all parameters in this model are learned end-to-end. We apply our framework to multi-view clustering and semi-supervised classification tasks to learn a latent compact representation. Experimental results demonstrate the superiority of the proposed framework over state-of-the-art multi-view learning models.

Despite the success of stochastic variance-reduced gradient (SVRG) algorithms in solving large-scale problems, their stochastic gradient complexity often scales linearly with data size and is expensive for huge data. Accordingly, we propose a hybrid stochastic-deterministic minibatch proximal gradient (HSDMPG) algorithm for strongly convex problems with linear prediction structure, e.g., least squares and logistic/softmax regression.HSDMPGenjoys improved computational complexity that is data-size-independent for large-scale problems. It iteratively samples an evolving minibatch of individual losses to estimate the original problem, and can efficiently minimize the sampled subproblems. For strongly convex loss of$n$components,HSDMPGattains an$\epsilon$-optimization-error within$\mathcal {O} \left(\kappa \log ^{\zeta +1}\left(\frac{1}{\epsilon }\right)\frac{1}{\epsilon }\bigwedge n\log ^{\zeta }\left(\frac{1}{\epsilon }\right)\right)$stochastic gradient evaluations, where$\kappa$is condition number,$\zeta =1$for quadratic loss and$\zeta =2$for generic loss. For large-scale problems, our complexity outperforms those of SVRG-type algorithms with/without dependence on data size. Particularly, when$\epsilon =\mathcal {O}(1/\sqrt{n})$which matches the intrinsic excess error of a learning model and is sufficient for generalization, our complexity for quadratic and generic losses is respectively$\mathcal {O} (n^{0.5}\log ^{2}(n))$and$\mathcal {O} (n^{0.5}\log ^{3}(n))$, which for the first time achieves optimal generalization in less than a single pass over data. Besides, we extendHSDMPGto online strongly convex problems and prove its higher efficiency over the prior algorithms. Numerical results demonstrate the computational advantages ofHSDMPG.

Recently, the study on learned iterative shrinkage thresholding algorithm (LISTA) has attracted increasing attentions. A large number of experiments as well as some theories have proved the high efficiency of LISTA for solving sparse coding problems. However, existing LISTA methods are all serial connection. To address this issue, we propose a novel extragradient based LISTA (ELISTA), which has a residual structure and theoretical guarantees. Moreover, most LISTA methods use the soft thresholding function, which has been found to cause a large estimation bias. Therefore, we propose a thresholding function for ELISTA instead of soft thresholding. From a theoretical perspective, we prove that our method attains linear convergence. Through ablation experiments, the improvements of our method on the network structure and the thresholding function are verified in practice. Extensive empirical results verify the advantages of our method.

Spherical signals exist in many applications, e.g., planetary data, LiDAR scans and digitalization of 3D objects, calling for models that can process spherical data effectively. It does not perform well when simply projecting spherical data into the 2D plane and then using planar convolution neural networks (CNNs), because of the distortion from projection and ineffective translation equivariance. Actually, good principles of designing spherical CNNs are avoiding distortions and converting the shift equivariance property in planar CNNs to rotation equivariance in the spherical domain. In this work, we use partial differential operators (PDOs) to design a spherical equivariant CNN, PDO-eS2CNN, which is exactly rotation equivariant in the continuous domain. We then discretize PDO-eS2CNNs, and analyze the equivariance error resulted from discretization. This is the first time that the equivariance error is theoretically analyzed in the spherical domain. In experiments, PDO-eS2CNNs show greater parameter efficiency and outperform other spherical CNNs significantly on several tasks.

Aerial Image Segmentation is a particular semantic segmentation problem and has several challenging characteristics that general semantic segmentation does not have. There are two critical issues: The one is an extremely foreground-background imbalanced distribution, and the other is multiple small objects along with the complex background. Such problems make the recent dense affinity context modeling perform poorly even compared with baselines due to over-introduced background context. To handle these problems, we propose a point-wise affinity propagation module based on the Feature Pyramid Network (FPN) framework, named PointFlow. Rather than dense affinity learning, a sparse affinity map is generated upon selected points between the adjacent features, which reduces the noise introduced by the background while keeping efficiency. In particular, we design a dual point matcher to select points from the salient area and object boundaries, respectively. Experimental results on three different aerial segmentation datasets suggest that the proposed method is more effective and efficient than state-of-the-art general semantic segmentation methods. Especially, our methods achieve the best speed and accuracy trade-off on three aerial benchmarks. Further experiments on three general semantic segmentation datasets prove the generality of our method. Code and models are made available (https://github.com/lxtGH/PFSegNets).

Most differentiable neural architecture search methods construct a super-net for search and derive a target-net as its sub-graph for evaluation. There exists a significant gap between the architectures in search and evaluation. As a result, current methods suffer from an inconsistent, inefficient, and inflexible search process. In this paper, we introduce EnTranNAS that is composed of Engine-cells and Transit-cells. The Engine-cell is differentiable for architecture search, while the Transit-cell only transits a sub-graph by architecture derivation. Consequently, the gap between the architectures in search and evaluation is significantly reduced. Our method also spares much memory and computation cost, which speeds up the search process. A feature sharing strategy is introduced for more balanced optimization and more efficient search. Furthermore, we develop an architecture derivation method to replace the traditional one that is based on a hand-crafted rule. Our method enables differentiable sparsification, and keeps the derived architecture equivalent to that of Engine-cell, which further improves the consistency between search and evaluation. More importantly, it supports the search for topology where a node can be connected to prior nodes with any number of connections, so that the searched architectures could be more flexible. Our search on CIFAR-10 has an error rate of 2.22% with only 0.07 GPU-day. We can also directly perform the search on ImageNet with topology learnable and achieve a top-1 error rate of 23.8% in 2.1 GPU-day.

Recently, some contrastive learning methods have been proposed to simultaneously learn representations and clustering assignments, achieving significant improvements. However, these methods do not take the category information and clustering objective into consideration, thus the learned representations are not optimal for clustering and the performance might be limited. Towards this issue, we first propose a novel graph contrastive learning framework, and then apply it to the clustering task, resulting in the Graph Constrastive Clustering (GCC) method. Different from basic contrastive clustering that only assumes an image and its augmentation should share similar representation and clustering assignments, we lift the instance-level consistency to the cluster-level consistency with the assumption that samples in one cluster and their augmentations should all be similar. Specifically, on the one hand, we propose the graph Laplacian based contrastive loss to learn more discriminative and clustering-friendly features. On the other hand, we propose a novel graph-based contrastive learning strategy to learn more compact clustering assignments. Both of them incorporate the latent category information to reduce the intra-cluster variance as well as increase the inter-cluster variance. Experiments on six commonly used datasets demonstrate the superiority of our proposed approach over the state-of-the-art methods.1

In this paper, we propose a density estimation algorithm called \textit{Gradient Boosting Histogram Transform} (GBHT), where we adopt the \textit{Negative Log Likelihood} as the loss function to make the boosting procedure available for the unsupervised tasks. From a learning theory viewpoint, we first prove fast convergence rates for GBHT with the smoothness assumption that the underlying density function lies in the space $C^{0,\alpha}$. Then when the target density function lies in spaces $C^{1,\alpha}$, we present an upper bound for GBHT which is smaller than the lower bound of its corresponding base learner, in the sense of convergence rates. To the best of our knowledge, we make the first attempt to theoretically explain why boosting can enhance the performance of its base learners for density estimation problems. In experiments, we not only conduct performance comparisons with the widely used KDE, but also apply GBHT to anomaly detection to showcase a further application of GBHT.

The compelling synthesis results of Generative Adversarial Networks (GANs) demonstrate rich semantic knowledge in their latent codes. To obtain this knowledge for downstream applications, encoding GANs has been proposed to learn encoders, such that real world data can be encoded to latent codes, which can be fed to generators to reconstruct those data. However, despite the theoretical guarantees of precise reconstruction in previous works, current algorithms generally reconstruct inputs with non-negligible deviations from inputs. In this paper we study this predicament of encoding GANs, which is indispensable research for the GAN community. We prove three uncertainty principles of encoding GANs in practice: a) the ‘perfect’ encoder and generator cannot be continuous at the same time, which implies that current framework of encoding GANs is ill-posed and needs rethinking; b) neural networks cannot approximate the underlying encoder and generator precisely at the same time, which explains why we cannot get ‘perfect’ encoders and generators as promised in previous theories; c) neural networks cannot be stable and accurate at the same time, which demonstrates the difficulty of training and trade-off between fidelity and disentanglement encountered in previous works. Our work may eliminate gaps between previous theories and empirical results, promote the understanding of GANs, and guide network designs for follow-up works.

As an important branch of weakly supervised learning, partial label learning deals with data where each instance is assigned with a set of candidate labels, whereas only one of them is true. Despite many methodology studies on learning from partial labels, there still lacks theoretical understandings of their risk consistent properties under relatively weak assumptions, especially on the link between theoretical results and the empirical choice of parameters. In this paper, we propose a family of loss functions named \textit{Leveraged Weighted} (LW) loss, which for the first time introduces the leverage parameter $\beta$ to consider the trade-off between losses on partial labels and non-partial ones. From the theoretical side, we derive a generalized result of risk consistency for the LW loss in learning from partial labels, based on which we provide guidance to the choice of the leverage parameter $\beta$. In experiments, we verify the theoretical guidance, and show the high effectiveness of our proposed LW loss on both benchmark and real datasets compared with other state-of-the-art partial label learning algorithms.

Epilepsy is a chronic neurological disorder that affects many people in the world. Automatic epileptic detection based on multi-channel electroencephalogram (EEG) signals is of great significance and has been widely studied. Recent deep learning models fail to consider the weights of different EEG channels since a few channels can play more important roles than other ones. In this paper, we propose an end-to-end epilepsy detection model, CW-SRNet, to solve this problem. We design a novel channel-weighted block (CW-Block) to capture the different importance of EEG channels automatically and dynamically. We combine our novel CW-Block with the squeeze-and-excitation residual network to improve epilepsy detection performance. Experiments on two public EEG datasets show that our model achieves state-of-the-art performance. Particularly on the CHB-MIT dataset, our model achieves an average sensitivity of 96.84% and an average specificity of 99.68%, outperforming other methods with clear margins.

This paper focuses on training implicit models of infinite layers. Specifically, previous works employ implicit differentiation and solve the exact gradient for the backward propagation. However, is it necessary to compute such an exact but expensive gradient for training? In this work, we propose a novel gradient estimate for implicit models, named phantom gradient, that 1) forgoes the costly computation of the exact gradient; and 2) provides an update direction empirically preferable to the implicit model training. We theoretically analyze the condition under which an ascent direction of the loss landscape could be found and provide two specific instantiations of the phantom gradient based on the damped unrolling and Neumann series. Experiments on large-scale tasks demonstrate that these lightweight phantom gradients significantly accelerate the backward passes in training implicit models by roughly 1.7 $\times$ and even boost the performance over approaches based on the exact gradient on ImageNet.

Convolutional neural networks (CNNs) have dominated the field of Computer Vision and achieved great success due to their built-in translation equivariance. Group equivariant CNNs (G-CNNs) that incorporate more equivariance can significantly improve the performance of conventional CNNs. However, G-CNNs are faced with two major challenges: \emph{spatial-agnostic problem} and \emph{expensive computational cost}. In this work, we propose a general framework of previous equivariant models, which includes G-CNNs and equivariant self-attention layers as special cases. Under this framework, we explicitly decompose the feature aggregation operation into a kernel generator and an encoder, and decouple the spatial and extra geometric dimensions in the computation. Therefore, our filters are essentially dynamic rather than being spatial-agnostic. We further show that our \emph{E}quivariant model is parameter \emph{E}fficient and computation \emph{E}fficient by complexity analysis, and also data \emph{E}fficient by experiments, so we call our model $E^4$-Net. Extensive experiments verify that our model can significantly improve previous works with smaller model size.Especially, under the setting of training on $1/5$ data of CIFAR10, our model improves G-CNNs by $5\%+$ accuracy,while using only $56\%$ parameters and $68\%$ FLOPs.

Attention mechanism has shown great performance and efficiency in a lot of deep learning models, in which relative position encoding plays a crucial role. However, when introducing attention to manifolds, there is no canonical local coordinate system to parameterize neighborhoods. To address this issue, we propose an equivariant transformer to make our model agnostic to the orientation of local coordinate systems (\textit{i.e.}, gauge equivariant), which employs multi-head self-attention to jointly incorporate both position-based and content-based information. To enhance expressive ability, we adopt regular field of cyclic groups as feature fields in intermediate layers, and propose a novel method to parallel transport the feature vectors in these fields. In addition, we project the position vector of each point onto its local coordinate system to disentangle the orientation of the coordinate system in ambient space (\textit{i.e.}, global coordinate system), achieving rotation invariance. To the best of our knowledge, we are the first to introduce gauge equivariance to self-attention, thus name our model Gauge Equivariant Transformer (GET), which can be efficiently implemented on triangle meshes. Extensive experiments show that GET achieves state-of-the-art performance on two common recognition tasks.

Multi-view methods learn representations by aligning multiple views of the same image and their performance largely depends on the choice of data augmentation. In this paper, we notice that some other useful augmentations, such as image rotation, are harmful for multi-view methods because they cause a semantic shift that is too large to be aligned well. This observation motivates us to relax the exact alignment objective to better cultivate stronger augmentations. Taking image rotation as a case study, we develop a generic approach, Pretext-aware Residual Relaxation (Prelax), that relaxes the exact alignment by allowing an adaptive residual vector between different views and encoding the semantic shift through pretext-aware learning. Extensive experiments on different backbones show that our method can not only improve multi-view methods with existing augmentations, but also benefit from stronger image augmentations like rotation.

Graph Convolutional Networks (GCNs) have attracted more and more attentions in recent years. A typical GCN layer consists of a linear feature propagation step and a nonlinear transformation step. Recent works show that a linear GCN can achieve comparable performance to the original non-linear GCN while being much more computationally efficient. In this paper, we dissect the feature propagation steps of linear GCNs from a perspective of continuous graph diffusion, and analyze why linear GCNs fail to benefit from more propagation steps. Following that, we propose Decoupled Graph Convolution (DGC) that decouples the terminal time and the feature propagation steps, making it more flexible and capable of exploiting a very large number of feature propagation steps. Experiments demonstrate that our proposed DGC improves linear GCNs by a large margin and makes them competitive with many modern variants of non-linear GCNs.

Spiking neural networks (SNNs) are brain-inspired models that enable energy-efficient implementation on neuromorphic hardware. However, the supervised training of SNNs remains a hard problem due to the discontinuity of the spiking neuron model. Most existing methods imitate the backpropagation framework and feedforward architectures for artificial neural networks, and use surrogate derivatives or compute gradients with respect to the spiking time to deal with the problem. These approaches either accumulate approximation errors or only propagate information limitedly through existing spikes, and usually require information propagation along time steps with large memory costs and biological implausibility. In this work, we consider feedback spiking neural networks, which are more brain-like, and propose a novel training method that does not rely on the exact reverse of the forward computation. First, we show that the average firing rates of SNNs with feedback connections would gradually evolve to an equilibrium state along time, which follows a fixed-point equation. Then by viewing the forward computation of feedback SNNs as a black-box solver for this equation, and leveraging the implicit differentiation on the equation, we can compute the gradient for parameters without considering the exact forward procedure. In this way, the forward and backward procedures are decoupled and therefore the problem of non-differentiable spiking functions is avoided. We also briefly discuss the biological plausibility of implicit differentiation, which only requires computing another equilibrium. Extensive experiments on MNIST, Fashion-MNIST, N-MNIST, CIFAR-10, and CIFAR-100 demonstrate the superior performance of our method for feedback models with fewer neurons and parameters in a small number of time steps. Our code is available at https://github.com/pkuxmq/IDE-FSNN.

Hashing, which represents data items as compact binary codes, has been becoming a more and more popular technique, e.g., for large-scale image retrieval, owing to its super fast search speed as well as its extremely economical memory consumption. However, existing hashing methods all try to learn binary codes from artificially balanced datasets which are not commonly available in real-world scenarios. In this paper, we propose Long-Tail Hashing Network (LTHNet), a novel two-stage deep hashing approach that addresses the problem of learning to hash for more realistic datasets where the data labels roughly exhibit a long-tail distribution. Specifically, the first stage is to learn relaxed embeddings of the given dataset with its long-tail characteristic taken into account via an end-to-end deep neural network; the second stage is to binarize those obtained embeddings. A critical part of LTHNet is its dynamic meta-embedding module extended with a determinantal point process which can adaptively realize visual knowledge transfer between head and tail classes, and thus enrich image representations for hashing. Our experiments have shown that LTHNet achieves dramatic performance improvements over all state-of-the-art competitors on long-tail datasets, with no or little sacrifice on balanced datasets. Further analyses reveal that while to our surprise directly manipulating class weights in the loss function has little effect, the extended dynamic meta-embedding module, the usage of cross-entropy loss instead of square loss, and the relatively small batch-size for training all contribute to LTHNet's success.

Abstract In recent years, convolutional neural networks have achieved great success in streak artifacts reduction. However, there is no special method designed for the artifacts reduction of the prostate. To solve the problem, the artifacts reduction CliqueNet (ARCliqueNet) to reconstruct dense‐view computed tomography images form sparse‐view computed tomography images is proposed. In detail, first, the proposed ARCliqueNet extracts a set of feature maps from the prostate sparse‐view CT image by Clique Block. Second, the feature maps are sent to ASPP with memory to be refined. Thenanother Clique Block is applied to the output of ASPP with memory and reconstruct the dense‐view CT images. Later on, reconstructed dense‐view CT images are used as new input of the original network. This process is repeated recurrently with memory delivering information between these recurrent stages. The final reconstructed dense‐view CT images are the output of the last recurrent stage. Our proposed ARCliqueNet outperforms the SOTA (state‐of‐the‐art) general artifacts reduction methods on the prostate dataset in terms of PSNR (peak signal‐to‐noise ratio) and SSIM (structural similarity). Therefore, we can draw the conclusion that Clique structures, ASPP with memory and recurrent learning are useful for prostate sparse‐view CT Artifacts here.

In this paper, we propose a novel algorithm for large-scale regression problems named Histogram Transform Ensembles (HTE), composed of random rotations, stretchings, and translations. Our HTE method first implements a histogram transformed partition to the random affine mapped data, then adaptively leverages constant functions or SVMs to obtain the individual regression estimates, and eventually builds the ensemble predictor through an average strategy. First of all, in this paper, we investigate the theoretical properties of HTE when the regression function lies in the Hölder space $C^{k,\alpha}$, $k \in \mathbb{N}_0$, $\alpha \in (0,1]$. In the case that $k=0, 1$, we adopt the constant regressors and develop the naïve histogram transforms (NHT). Within the space $C^{0,\alpha}$, although almost optimal convergence rates can be derived for both single and ensemble NHT, we fail to show the benefits of ensembles over single estimators theoretically. In contrast, in the subspace $C^{1,\alpha}$, we prove that if $d \geq 2(1+\alpha)/\alpha$, the lower bound of the convergence rates for single NHT turns out to be worse than the upper bound of the convergence rates for ensemble NHT. In the other case when $k \geq 2$, the NHT may no longer be appropriate in predicting smoother regression functions. Instead, we circumvent this issue by applying kernel histogram transforms (KHT) equipped with smoother regressors, such as support vector machines (SVMs). Accordingly, it turns out that both single and ensemble KHT enjoy almost optimal convergence rates. Then, we validate the above theoretical results with extensive numerical experiments. On the one hand, simulations are conducted to elucidate that ensemble NHT outperforms single NHT. On the other hand, the effects of bin sizes on the accuracy of both NHT and KHT are also in accord with the theoretical analysis. Last but not least, in the real-data experiments, comparisons between the ensemble KHT, equipped with adaptive histogram transforms, and other state-of-the-art large-scale regression estimators verify the effectiveness and precision of the proposed algorithm.

Recently, many stochastic variance reduced alternating direction methods of multipliers (ADMMs) (e.g., SAG-ADMM and SVRG-ADMM) have made exciting progress such as linear convergence rate for strongly convex (SC) problems. However, their best-known convergence rate for non-strongly convex (non-SC) problems is$\mathcal {O}(1/T)$as opposed to$\mathcal {O}(1/T^2)$of accelerated deterministic algorithms, where$T$is the number of iterations. Thus, there remains a gap in the convergence rates of existing stochastic ADMM and deterministic algorithms. To bridge this gap, we introduce a new momentum acceleration trick into stochastic variance reduced ADMM, and propose a novel accelerated SVRG-ADMM method (called ASVRG-ADMM) for the machine learning problems with the constraint$Ax + By = c$. Then we design a linearized proximal update rule and a simple proximal one for the two classes of ADMM-style problems with$B = \tau I$and$B\ne \tau I$, respectively, where$I$is an identity matrix and$\tau$is an arbitrary bounded constant. Note that our linearized proximal update rule can avoid solving sub-problems iteratively. Moreover, we prove that ASVRG-ADMM converges linearly for SC problems. In particular, ASVRG-ADMM improves the convergence rate from$\mathcal {O}(1/T)$to$\mathcal {O}(1/T^2)$for non-SC problems. Finally, we apply ASVRG-ADMM to various machine learning problems, e.g., graph-guided fused Lasso, graph-guided logistic regression, graph-guided SVM, generalized graph-guided fused Lasso and multi-task learning, and show that ASVRG-ADMM consistently converges faster than the state-of-the-art methods.

Neural architecture search (NAS) is inherently subject to the gap of architectures during searching and validating. To bridge this gap effectively, we develop Differentiable ArchiTecture Approximation (DATA) with Ensemble Gumbel-Softmax (EGS) estimator and Architecture Distribution Constraint (ADC) to automatically approximate architectures during searching and validating in a differentiable manner. Technically, the EGS estimator consists of a group of Gumbel-Softmax estimators, which is capable of converting probability vectors to binary codes and passing gradients reversely, reducing the estimation bias in a differentiable way. To narrow the distribution gap between sampled architectures and supernet, further, the ADC is introduced to reduce the variance of sampling during searching. Benefiting from such modeling, architecture probabilities and network weights in the NAS model can be jointly optimized with the standard back-propagation, yielding an end-to-end learning mechanism for searching deep neural architectures in an extended search space. Conclusively, in the validating process, a high-performance architecture that approaches to the learned one during searching is readily built. Extensive experiments on various tasks including image classification, few-shot learning, unsupervised clustering, semantic segmentation and language modeling strongly demonstrate that DATA is capable of discovering high-performance architectures while guaranteeing the required efficiency. Code is available at https://github.com/XinbangZhang/DATA-NAS.

Multi-way or tensor data analysis has attracted increasing attention recently, with many important applications in practice. This article develops a tensor low-rank representation (TLRR) method, which is the first approach that can exactly recover the clean data of intrinsic low-rank structure and accurately cluster them as well, with provable performance guarantees. In particular, for tensor data with arbitrary sparse corruptions, TLRR can exactly recover the clean data under mild conditions; meanwhile TLRR can exactly verify their true origin tensor subspaces and hence cluster them accurately. TLRR objective function can be optimized via efficient convex programing with convergence guarantees. Besides, we provide two simple yet effective dictionary construction methods, the simple TLRR (S-TLRR) and robust TLRR (R-TLRR), to handle slightly and severely corrupted data respectively. Experimental results on two computer vision data analysis tasks, image/video recovery and face clustering, clearly demonstrate the superior performance, efficiency and robustness of our developed method over state-of-the-arts including the popular LRR and SSC methods.

Graph-based convolutional model such as non-local block has shown to be effective for strengthening the context modeling ability in convolutional neural networks (CNNs). However, its pixel-wise computational overhead is prohibitive which renders it unsuitable for high resolution imagery. In this paper, we explore the efficiency of context graph reasoning and propose a novel framework called Squeeze Reasoning. Instead of propagating information on the spatial map, we first learn to squeeze the input feature into a channel-wise global vector and perform reasoning within the single vector where the computation cost can be significantly reduced. Specifically, we build the node graph in the vector where each node represents an abstract semantic concept. The refined feature within the same semantic category results to be consistent, which is thus beneficial for downstream tasks. We show that our approach can be modularized as an end-to-end trained block and can be easily plugged into existing networks. Despite its simplicity and being lightweight, the proposed strategy allows us to establish the considerable results on different semantic segmentation datasets and shows significant improvements with respect to strong baselines on various other scene understanding tasks including object detection, instance segmentation and panoptic segmentation. Code is available at https://github.com/lxtGH/SFSegNets.

Monocular object pose estimation is an important yet challenging computer vision problem. Depth features can provide useful information for pose estimation. However, existing methods rely on real depth images to extract depth features, leading to its difficulty on various applications. In this paper, we aim at extracting RGB and depth features from a single RGB image with the help of synthetic RGB-depth image pairs for object pose estimation. Specifically, a deep convolutional neural network is proposed with an RGB-to-Depth Embedding module and a Synthetic-Real Adaptation module. The embedding module is trained with synthetic pair data to learn a depth-oriented embedding space between RGB and depth images optimized for object pose estimation. The adaptation module is to further align distributions from synthetic to real data. Compared to existing methods, our method does not need any real depth images and can be trained easily with large-scale synthetic data. Extensive experiments and comparisons show that our method achieves best performance on a challenging public PASCAL 3D+ dataset in all the metrics, which substantiates the superiority of our method and the above modules.

Graph Convolutional Networks (GCNs) play a crucial role in graph learning tasks, however, learning graph embedding with few supervised signals is still a difficult problem. In this paper, we propose a novel training algorithm for Graph Convolutional Network, called Multi-Stage Self-Supervised (M3S) Training Algorithm, combined with self-supervised learning approach, focusing on improving the generalization performance of GCNs on graphs with few labeled nodes. Firstly, a Multi-Stage Training Framework is provided as the basis of M3S training method. Then we leverage DeepCluster technique, a popular form of self-supervised learning, and design corresponding aligning mechanism on the embedding space to refine the Multi-Stage Training Framework, resulting in M3S Training Algorithm. Finally, extensive experimental results verify the superior performance of our algorithm on graphs with few labeled nodes under different label rates compared with other state-of-the-art approaches.

Multi-view clustering aims to take advantage of multiple views information to improve the performance of clustering. Many existing methods compute the affinity matrix by low-rank representation (LRR) and pairwise investigate the relationship between views. However, LRR suffers from the high computational cost in self-representation optimization. Besides, compared with pairwise views, tensor form of all views' representation is more suitable for capturing the high-order correlations among all views. Towards these two issues, in this paper, we propose the unified graph and low-rank tensor learning (UGLTL) for multi-view clustering. Specifically, on the one hand, we learn the view-specific affinity matrix based on projected graph learning. On the other hand, we reorganize the affinity matrices into tensor form and learn its intrinsic tensor based on low-rank tensor approximation. Finally, we unify these two terms together and jointly learn the optimal projection matrices, affinity matrices and intrinsic low-rank tensor. We also propose an efficient algorithm to iteratively optimize the proposed model. To evaluate the performance of the proposed method, we conduct extensive experiments on multiple benchmarks across different scenarios and sizes. Compared with the state-of-the-art approaches, our method achieves much better performance.

The panoptic segmentation task requires a unified result from semantic and instance segmentation outputs that may contain overlaps. However, current studies widely ignore modeling overlaps. In this study, we aim to model overlap relations among instances and resolve them for panoptic segmentation. Inspired by scene graph representation, we formulate the overlapping problem as a simplified case, named scene overlap graph. We leverage each object's category, geometry and appearance features to perform relational embedding, and output a relation matrix that encodes overlap relations. In order to overcome the lack of supervision, we introduce a differentiable module to resolve the overlap between any pair of instances. The mask logits after removing overlaps are fed into per-pixel instance id classification, which leverages the panoptic supervision to assist in the modeling of overlap relations. Besides, we generate an approximate ground truth of overlap relations as the weak supervision, to quantify the accuracy of overlap relations predicted by our method. Experiments on COCO and Cityscapes demonstrate that our method is able to accurately predict overlap relations, and outperform the state-of-the-art performance for panoptic segmentation. Our method also won the Innovation Award in COCO 2019 challenge.

The correspondence between residual networks and dynamical systems motivates researchers to unravel the physics of ResNets with well-developed tools in numeral methods of ODE systems. The Runge-Kutta-Fehlberg method is an adaptive time stepping that renders a good trade-off between the stability and efficiency. Can we also have an adaptive time stepping for ResNets to ensure both stability and performance? In this study, we analyze the effects of time stepping on the Euler method and ResNets. We establish a stability condition for ResNets with step sizes and weight parameters, and point out the effects of step sizes on the stability and performance. Inspired by our analyses, we develop an adaptive time stepping controller that is dependent on the parameters of the current step, and aware of previous steps. The controller is jointly optimized with the network training so that variable step sizes and evolution time can be adaptively adjusted. We conduct experiments on ImageNet and CIFAR to demonstrate the effectiveness. It is shown that our proposed method is able to improve both stability and accuracy without introducing additional overhead in inference phase.

The convolution operation suffers from a limited receptive filed, while global modeling is fundamental to dense prediction tasks, such as semantic segmentation. In this paper, we apply graph convolution into the semantic segmentation task and propose an improved Laplacian. The graph reasoning is directly performed in the original feature space organized as a spatial pyramid. Different from existing methods, our Laplacian is data-dependent and we introduce an attention diagonal matrix to learn a better distance metric. It gets rid of projecting and re-projecting processes, which makes our proposed method a light-weight module that can be easily plugged into current computer vision architectures. More importantly, performing graph reasoning directly in the feature space retains spatial relationships and makes spatial pyramid possible to explore multiple long-range contextual patterns from different scales. Experiments on Cityscapes, COCO Stuff, PASCAL Context and PASCAL VOC demonstrate the effectiveness of our proposed methods on semantic segmentation. We achieve comparable performance with advantages in computational and memory overhead.

In this paper, we propose a boosting algorithm for regression problems called \emph{boosted histogram transform for regression} (BHTR) based on histogram transforms composed of random rotations, stretchings, and translations. From the theoretical perspective, we first prove fast convergence rates for BHTR under the assumption that the target function lies in the spaces $C^{0,\alpha}$. Moreover, if the target function resides in the subspace $C^{1,\alpha}$, by establishing the upper bound of the convergence rate for the boosted regressor, i.e. BHTR, and the lower bound for base regressors, i.e. histogram transform regressors (HTR), we manage to explain the benefits of the boosting procedure. In the experiments, compared with other state-of-the-art algorithms such as gradient boosted regression tree (GBRT), Breiman’s forest, and kernel-based methods, our BHTR algorithm shows promising performance on both synthetic and real datasets.

Deep neural networks have achieved great success in various areas, but recent works have found that neural networks are vulnerable to adversarial attacks, which leads to a hot topic nowadays. Although many approaches have been proposed to enhance the robustness of neural networks, few of them explored robust architectures for neural networks. On this account, we try to address such an issue from the perspective of dynamic system in this work. By viewing ResNet as an explicit Euler discretization of an ordinary differential equation (ODE), for the first time, we find that the adversarial robustness of ResNet is connected to the numerical stability of the corresponding dynamic system, i.e., more stable numerical schemes may correspond to more robust deep networks. Furthermore, inspired by the implicit Euler method for solving numerical ODE problems, we propose Implicit Euler skip connections (IE-Skips) by modifying the original skip connection in ResNet or its variants. Then we theoretically prove its advantages under the adversarial attack and the experimental results show that our ResNet with IE-Skips can largely improve the robustness and the generalization ability under adversarial attacks when compared with the vanilla ResNet of the same parameter size.

Recent research has shown that incorporating equivariance into neural network architectures is very helpful, and there have been some works investigating the equivariance of networks under group actions. However, as digital images and feature maps are on the discrete meshgrid, corresponding equivariance-preserving transformation groups are very limited. In this work, we deal with this issue from the connection between convolutions and partial differential operators (PDOs). In theory, assuming inputs to be smooth, we transform PDOs and propose a system which is equivariant to a much more general continuous group, the $n$-dimension Euclidean group. In implementation, we discretize the system using the numerical schemes of PDOs, deriving approximately equivariant convolutions (PDO-eConvs). Theoretically, the approximation error of PDO-eConvs is of the quadratic order. It is the first time that the error analysis is provided when the equivariance is approximate. Extensive experiments on rotated MNIST and natural image classification show that PDO-eConvs perform competitively yet use parameters much more efficiently. Particularly, compared with Wide ResNets, our methods result in better results using only 12.6% parameters.

While successful in many fields, deep neural networks (DNNs) still suffer from some open problems such as bad local minima and unsatisfactory generalization performance. In this work, we propose a novel architecture called Maximum-and-Concatenation Networks (MCN) to try eliminating bad local minima and improving generalization ability as well. Remarkably, we prove that MCN has a very nice property; that is, every local minimum of an (l+1)-layer MCN can be better than, at least as good as, the global minima of the network consisting of its first l layers. In other words, by increasing the network depth, MCN can autonomously improve its local minima’s goodness, what is more, it is easy to plug MCN into an existing deep model to make it also have this property. Finally, under mild conditions, we show that MCN can approximate certain continuous function arbitrarily well with high efficiency; that is, the covering number of MCN is much smaller than most existing DNNs such as deep ReLU. Based on this, we further provide a tight generalization bound to guarantee the inference ability of MCN when dealing with testing samples.

Neural architecture search (NAS) aims to produce the optimal sparse solution from a high-dimensional space spanned by all candidate connections. Current gradient-based NAS methods commonly ignore the constraint of sparsity in the search phase, but project the optimized solution onto a sparse one by post-processing. As a result, the dense super-net for search is inefficient to train and has a gap with the projected architecture for evaluation. In this paper, we formulate neural architecture search as a sparse coding problem. We perform the differentiable search on a compressed lower-dimensional space that has the same validation loss as the original sparse solution space, and recover an architecture by solving the sparse coding problem. The differentiable search and architecture recovery are optimized in an alternate manner. By doing so, our network for search at each update satisfies the sparsity constraint and is efficient to train. In order to also eliminate the depth and width gap between the network in search and the target-net in evaluation, we further propose a method to search and evaluate in one stage under the target-net settings. When training finishes, architecture variables are absorbed into network weights. Thus we get the searched architecture and optimized parameters in a single run. In experiments, our two-stage method on CIFAR-10 requires only 0.05 GPU-day for search. Our one-stage method produces state-of-the-art performances on both CIFAR-10 and ImageNet at the cost of only evaluation time.

Dual first-order methods are essential techniques for large-scale constrained convex optimization. However, when recovering the primal solutions, we need $T(\epsilon^{-2})$ iterations to achieve an $\epsilon$-optimal primal solution when we apply an algorithm to the non-strongly convex dual problem with $T(\epsilon^{-1})$ iterations to achieve an $\epsilon$-optimal dual solution, where $T(x)$ can be $x$ or $\sqrt{x}$. In this paper, we prove that the iteration complexity of the primal solutions and dual solutions have the same $O\left(\frac{1}{\sqrt{\epsilon}}\right)$ order of magnitude for the accelerated randomized dual coordinate ascent. When the dual function further satisfies the quadratic functional growth condition, by restarting the algorithm at any period, we establish the linear iteration complexity for both the primal solutions and dual solutions even if the condition number is unknown. When applied to the regularized empirical risk minimization problem, we prove the iteration complexity of $O\left(n\log n+\sqrt{\frac{n}{\epsilon}}\right)$ in both primal space and dual space, where $n$ is the number of samples. Our result takes out the $\left(\log \frac{1}{\epsilon}\right)$ factor compared with the methods based on smoothing/regularization or Catalyst reduction. As far as we know, this is the first time that the optimal $O\left(\sqrt{\frac{n}{\epsilon}}\right)$ iteration complexity in the primal space is established for the dual coordinate ascent based stochastic algorithms. We also establish the accelerated linear complexity for some problems with nonsmooth loss, e.g., the least absolute deviation and SVM.

Numerous tasks at the core of statistics, learning and vision areas are specific cases of ill-posed inverse problems. Recently, learning-based (e.g., deep) iterative methods have been empirically shown to be useful for these problems. Nevertheless, integrating learnable structures into iterations is still a laborious process, which can only be guided by intuitions or empirical insights. Moreover, there is a lack of rigorous analysis about the convergence behaviors of these reimplemented iterations, and thus the significance of such methods is a little bit vague. This paper moves beyond these limits and proposes Flexible Iterative Modularization Algorithm (FIMA), a generic and provable paradigm for nonconvex inverse problems. Our theoretical analysis reveals that FIMA allows us to generate globally convergent trajectories for learning-based iterative methods. Meanwhile, the devised scheduling policies on flexible modules should also be beneficial for classical numerical methods in the nonconvex scenario. Extensive experiments on real applications verify the superiority of FIMA.

In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product) [14]. Induced by the t-product, we first rigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor average rank, and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm. These definitions, their relationships and properties are consistent with matrix cases. Equipped with the new tensor nuclear norm, we then solve the TRPCA problem by solving a convex program and provide the theoretical guarantee for the exact recovery. Our TRPCA model and recovery guarantee include matrix RPCA as a special case. Numerical experiments verify our results, and the applications to image recovery and background modeling problems demonstrate the effectiveness of our method.

Numerical optimization serves as one of the pillars of machine learning. To meet the demands of big data applications, lots of efforts have been put on designing theoretically and practically fast algorithms. This article provides a comprehensive survey on accelerated first-order algorithms with a focus on stochastic algorithms. Specifically, this article starts with reviewing the basic accelerated algorithms on deterministic convex optimization, then concentrates on their extensions to stochastic convex optimization, and at last introduces some recent developments on acceleration for nonconvex optimization.

The prevailing characteristics of micro-videos result in the less descriptive power of each modality. The micro-video representations, several pioneer efforts proposed, are limited in implicitly exploring the consistency between different modality information but ignore the complementarity. In this paper, we focus on how to explicitly separate the consistent features and the complementary features from the mixed information and harness their combination to improve the expressiveness of each modality. Toward this end, we present a neural multimodal cooperative learning (NMCL) model to split the consistent component and the complementary component by a novel relation-aware attention mechanism. Specifically, the computed attention score can be used to measure the correlation between the features extracted from different modalities. Then, a threshold is learned for each modality to distinguish the consistent and complementary features according to the score. Thereafter, we integrate the consistent parts to enhance the representations and supplement the complementary ones to reinforce the information in each modality. As to the problem of redundant information, which may cause overfitting and is hard to distinguish, we devise an attention network to dynamically capture the features which closely related the category and output a discriminative representation for prediction. The experimental results on a real-world micro-video dataset show that the NMCL outperforms the state-of-the-art methods. Further studies verify the effectiveness and cooperative effects brought by the attentive mechanism.

We propose a new optimization method for training feedforward neural networks. By rewriting the activation function as an equivalent proximal operator, we approximate a feedforward neural network by adding the proximal operators to the objective function as penalties, hence we call the lifted proximal operator machine (LPOM). LPOM is block multiconvex in all layer-wise weights and activations. This allows us to use block coordinate descent to update the layer-wise weights and activations. Most notably, we only use the mapping of the activation function itself, rather than its derivative, thus avoiding the gradient vanishing or blow-up issues in gradient based training methods. So our method is applicable to various non-decreasing Lipschitz continuous activation functions, which can be saturating and non-differentiable. LPOM does not require more auxiliary variables than the layer-wise activations, thus using roughly the same amount of memory as stochastic gradient descent (SGD) does. Its parameter tuning is also much simpler. We further prove the convergence of updating the layer-wise weights and activations and point out that the optimization could be made parallel by asynchronous update. Experiments on MNIST and CIFAR-10 datasets testify to the advantages of LPOM.

In this paper, we give a sharp analysis for Stochastic Gradient Descent (SGD) and prove that SGD is able to efficiently escape from saddle points and find an $(\epsilon, O(\epsilon^{0.5}))$-approximate second-order stationary point in $\tilde{O}(\epsilon^{-3.5})$ stochastic gradient computations for generic nonconvex optimization problems, when the objective function satisfies gradient-Lipschitz, Hessian-Lipschitz, and dispersive noise assumptions. This result subverts the classical belief that SGD requires at least $O(\epsilon^{-4})$ stochastic gradient computations for obtaining an $(\epsilon,O(\epsilon^{0.5}))$-approximate second-order stationary point. Such SGD rate matches, up to a polylogarithmic factor of problem-dependent parameters, the rate of most accelerated nonconvex stochastic optimization algorithms that adopt additional techniques, such as Nesterov’s momentum acceleration, negative curvature search, as well as quadratic and cubic regularization tricks. Our novel analysis gives new insights into nonconvex SGD and can be potentially generalized to a broad class of stochastic optimization algorithms.

Subspace clustering methods based on data self-expression have become very popular for learning from data that lie in a union of low-dimensional linear subspaces. However, the applicability of subspace clustering has been limited because practical visual data in raw form do not necessarily lie in such linear subspaces. On the other hand, while Convolutional Neural Network (ConvNet) has been demonstrated to be a powerful tool for extracting discriminative features from visual data, training such a ConvNet usually requires a large amount of labeled data, which are unavailable in subspace clustering applications. To achieve simultaneous feature learning and subspace clustering, we propose an end-to-end trainable framework, called Self-Supervised Convolutional Subspace Clustering Network (S$^2$ConvSCN), that combines a ConvNet module (for feature learning), a self-expression module (for subspace clustering) and a spectral clustering module (for self-supervision) into a joint optimization framework. Particularly, we introduce a dual self-supervision that exploits the output of spectral clustering to supervise the training of the feature learning module (via a classification loss) and the self-expression module (via a spectral clustering loss). Our experiments on four benchmark datasets show the effectiveness of the dual self-supervision and demonstrate superior performance of our proposed approach.

Self-attention mechanism has been widely used for various tasks. It is designed to compute the representation of each position by a weighted sum of the features at all positions. Thus, it can capture long-range relations for computer vision tasks. However, it is computationally consuming. Since the attention maps are computed w.r.t all other positions. In this paper, we formulate the attention mechanism into an expectation-maximization manner and iteratively estimate a much more compact set of bases upon which the attention maps are computed. By a weighted summation upon these bases, the resulting representation is low-rank and deprecates noisy information from the input. The proposed Expectation-Maximization Attention (EMA) module is robust to the variance of input and is also friendly in memory and computation. Moreover, we set up the bases maintenance and normalization methods to stabilize its training procedure. We conduct extensive experiments on popular semantic segmentation benchmarks including PASCAL VOC, PASCAL Context, and COCO Stuff, on which we set new records1.

Recent developed deep unsupervised methods allow us to jointly learn representation and cluster unlabelled data. These deep clustering methods %like DAC start with mainly focus on the correlation among samples, e.g., selecting high precision pairs to gradually tune the feature representation, which neglects other useful correlations. In this paper, we propose a novel clustering framework, named deep comprehensive correlation mining~(DCCM), for exploring and taking full advantage of various kinds of correlations behind the unlabeled data from three aspects: 1) Instead of only using pair-wise information, pseudo-label supervision is proposed to investigate category information and learn discriminative features. 2) The features' robustness to image transformation of input space is fully explored, which benefits the network learning and significantly improves the performance. 3) The triplet mutual information among features is presented for clustering problem to lift the recently discovered instance-level deep mutual information to a triplet-level formation, which further helps to learn more discriminative features. Extensive experiments on several challenging datasets show that our method achieves good performance, e.g., attaining 62.3% clustering accuracy on CIFAR-10, which is 10.1% higher than the state-of-the-art results.

Recently, a number of learning-based optimization methods that combine data-driven architectures with the classical optimization algorithms have been proposed and explored, showing superior empirical performance in solving various ill-posed inverse problems, but there is still a scarcity of rigorous analysis about the convergence behaviors of learning-based optimization. In particular, most existing analyses are specific to unconstrained problems but cannot apply to the more general cases where some variables of interest are subject to certain constraints. In this paper, we propose Differentiable Linearized ADMM (D-LADMM) for solving the problems with linear constraints. Specifically, D-LADMM is a K-layer LADMM inspired deep neural network, which is obtained by firstly introducing some learnable weights in the classical Linearized ADMM algorithm and then generalizing the proximal operator to some learnable activation function. Notably, we rigorously prove that there exist a set of learnable parameters for D-LADMM to generate globally converged solutions, and we show that those desired parameters can be attained by training D-LADMM in a proper way. To the best of our knowledge, we are the first to provide the convergence analysis for the learning-based optimization method on constrained problems.

This paper studies the subspace clustering problem. Given some data points approximately drawn from a union of subspaces, the goal is to group these data points into their underlying subspaces. Many subspace clustering methods have been proposed and among which sparse subspace clustering and low-rank representation are two representative ones. Despite the different motivations, we observe that many existing methods own the common block diagonal property, which possibly leads to correct clustering, yet with their proofs given case by case. In this work, we consider a general formulation and provide a unified theoretical guarantee of the block diagonal property. The block diagonal property of many existing methods falls into our special case. Second, we observe that many existing methods approximate the block diagonal representation matrix by using different structure priors, e.g., sparsity and low-rankness, which are indirect. We propose the first block diagonal matrix induced regularizer for directly pursuing the block diagonal matrix. With this regularizer, we solve the subspace clustering problem by Block Diagonal Representation (BDR), which uses the block diagonal structure prior. The BDR model is nonconvex and we propose an alternating minimization solver and prove its convergence. Experiments on real datasets demonstrate the effectiveness of BDR.

We propose a method for estimating 3D human poses from single images or video sequences. The task is challenging because: (a) many 3D poses can have similar 2D pose projections which makes the lifting ambiguous, and (b) current 2D joint detectors are not accurate which can cause big errors in 3D estimates. We represent 3D poses by a sparse combination of bases which encode structural pose priors to reduce the lifting ambiguity. This prior is strengthened by adding limb length constraints. We estimate the 3D pose by minimizing an$L_1$norm measurement error between the 2D pose and the 3D pose because it is less sensitive to inaccurate 2D poses. We modify our algorithm to output$K$3D pose candidates for an image, and for videos, we impose a temporal smoothness constraint to select the best sequence of 3D poses from the candidates. We demonstrate good results on 3D pose estimation from static images and improved performance by selecting the best 3D pose from the$K$proposals. Our results on video sequences also show improvements (over static images) of roughly 15%.

In this paper, we study the subspace clustering problem under complex noise. A wide class of reconstruction-based methods model the subspace clustering problem by combining a quadratic data-fidelity term and a regularization term. In a statistical framework, the data-fidelity term assumes to be contaminated by a unimodal Gaussian noise, which is a popular setting in most current subspace clustering models. However, the realistic noise is much more complex than our assumptions. Besides, the coarse representation of the data-fidelity term may depress the clustering accuracy, which is often used to evaluate the models. To address this issue, we propose the mixture of Gaussian regression (MoG Regression) for subspace clustering. The MoG Regression seeks a valid way to model the unknown noise distribution, which approaches the real one as far as possible, so that the desired affinity matrix is better at characterizing the structure of data in the real world, and furthermore, improving the performance. Theoretically, the proposed model enjoys the grouping effect, which encourages the coefficients of highly correlated points are nearly equal. Drawing upon the ideal of the minimum message length, a model selection strategy is proposed to estimate the numbers of the Gaussian components that shows a way how to seek the number of Gaussian components besides determining it by empirical value. In addition, the asymptotic property of our model is investigated. The proposed model is evaluated on the challenging datasets. The experimental results show that the proposed MoG Regression model significantly outperforms several state-of-the-art subspace clustering methods.

Recently, multi-view clustering attracts much attention, which aims to take advantage of multi-view information to improve the performance of clustering. However, most recent work mainly focuses on the self-representation-based subspace clustering, which is of high computation complexity. In this paper, we focus on the Markov chain-based spectral clustering method and propose a novel essential tensor learning method to explore the high-order correlations for multi-view representation. We first construct a tensor based on multi-view transition probability matrices of the Markov chain. By incorporating the idea from the robust principle component analysis, tensor singular value decomposition (t-SVD)-based tensor nuclear norm is imposed to preserve the low-rank property of the essential tensor, which can well capture the principle information from multiple views. We also employ the tensor rotation operator for this task to better investigate the relationship among views as well as reduce the computation complexity. The proposed method can be efficiently optimized by the alternating direction method of multipliers (ADMM). Extensive experiments on seven real-world datasets corresponding to five different applications show that our method achieves superior performance over other state-of-the-art methods.

Mismatch removal is a key step in many computer vision problems that involve point matching. The existing methods for checking geometric consistency mainly focus on similarity or affine transformations. In this paper, we propose a novel mismatch removal method that can cope with the projective transformation between two corresponding point sets. Our approach is based on the augmented homogeneous coordinates matrix constructed from the coordinates of anchor matches, whose degeneracy can indicate the correctness of anchor matches. The set of anchor matches is initially all the matches and is iteratively updated by calculating the difference between the estimated matched points, which can be easily computed in a closed form, and the actually matched points and removing those with large differences. Experimental results on synthetic 2D point matching data sets and real image matching data sets verify that our method achieves the highest $F$ -score among all the methods under similarity, affine, and projective transformations with noises and outliers. Our method can also achieve faster speed than all other iterative methods. Those non-iterative methods with slight advantage in speed are not competitive in accuracy when compared with ours. We also show that the set of anchor matches is stable through the iteration and the computation time grows very slowly with respect to the number of matched points. When applied to mismatch removal in partial-duplicate image search, our method achieves the best retrieval precision, and its computing time is also highly competitive.

Fisher’s criterion is one of the most popular discriminant criteria for feature extraction. It is defined as the generalized Rayleigh quotient of the between-class scatter distance to the within-class scatter distance. Consequently, Fisher’s criterion does not take advantage of the discriminant information in the class covariance differences, and hence, its discriminant ability largely depends on the class mean differences. If the class mean distances are relatively large compared with the within-class scatter distance, Fisher’s criterion-based discriminant analysis methods may achieve a good discriminant performance. Otherwise, it may not deliver good results. Moreover, we observe that the between-class distance of Fisher’s criterion is based on the$\ell _{2}$-norm, which would be disadvantageous to separate the classes with smaller class mean distances. To overcome the drawback of Fisher’s criterion, in this paper, we first derive a new discriminant criterion, expressed as amixture of absolute generalized Rayleigh quotients, based on a Bayes error upper bound estimation, where mixture of Gaussians is adopted to approximate the real distribution of data samples. Then, the criterion is further modified by replacing$\ell _{2}$-norm with$\ell _{1}$one to better describe the between-class scatter distance, such that it would be more effective to separate the different classes. Moreover, we propose a novel$\ell _{1}$-norm heteroscedastic discriminant analysis method based on the new discriminant analysis (L1-HDA/GM) for heteroscedastic feature extraction, in which the optimization problem of L1-HDA/GM can be efficiently solved by using the eigenvalue decomposition approach. Finally, we conduct extensive experiments on four real data sets and demonstrate that the proposed method achieves much competitive results compared with the state-of-the-art methods.

Spectral Clustering (SC) is a widely used data clustering method which first learns a low-dimensional embedding U of data by computing the eigenvectors of the normalized Laplacian matrix, and then performs k-means on UT to get the final clustering result. The Sparse Spectral Clustering (SSC) method extends SC with a sparse regularization on UUT by using the block diagonal structure prior of UUT in the ideal case. However, encouraging UUT to be sparse leads to a heavily nonconvex problem which is challenging to solve and the work (Lu, Yan, and Lin 2016) proposes a convex relaxation in the pursuit of this aim indirectly. However, the convex relaxation generally leads to a loose approximation and the quality of the solution is not clear. This work instead considers to solve the nonconvex formulation of SSC which directly encourages UUT to be sparse. We propose an efficient Alternating Direction Method of Multipliers (ADMM) to solve the nonconvex SSC and provide the convergence guarantee. In particular, we prove that the sequences generated by ADMM always exist a limit point and any limit point is a stationary point. Our analysis does not impose any assumptions on the iterates and thus is practical. Our proposed ADMM for nonconvex problems allows the stepsize to be increasing but upper bounded, and this makes it very efficient in practice. Experimental analysis on several real data sets verifies the effectiveness of our method.

Highly incoherent dictionaries have broad applications in machine learning. Minimizing the mutual coherence is a common intuition to construct incoherent dictionaries in the previous methods. However, as pointed out by Tropp(2004), mutual coherence does not offer a very subtle description and Babel function, as a generalization of mutual coherence, is a more attractive alternative. However, it is much more challenging to optimize. In this work, we minimize the Babel function directly to construct incoherent dictionaries. As far as we know, this is the first work to optimize the Babel function. We propose an augmented Lagrange multiplier based algorithm to solve this nonconvex and nonsmooth problem with the convergence guarantee that every accumulation point is a KKT point. We define a new norm $\|\X\|_{\infty,max_p}$ and propose an efficient method to compute its proximal operation with $O(n^2\mbox{log}n)$ complexity, which dominates the running time of our algorithm, where $max_p$ means the sum of the largest $p$ elements and $n$ is the number of the atoms. Numerical experiments testify to the advantage of our method.

Deep neural networks have been one of the dominant machine learning approaches in recent years. Several new network structures are proposed and have better performance than the traditional feedforward neural network structure. Representative ones include the skip connection structure in ResNet and the dense connection structure in DenseNet. However, it still lacks a unified guidance for the neural network structure design. In this paper, we propose the hypothesis that the neural network structure design can be inspired by optimization algorithms and a faster optimization algorithm may lead to a better neural network structure. Specifically, we prove that the propagation in the feedforward neural network with the same linear transformation in different layers is equivalent to minimizing some function using the gradient descent algorithm. Based on this observation, we replace the gradient descent algorithm with the heavy ball algorithm and Nesterov’s accelerated gradient descent algorithm, which are faster and inspire us to design new and better network structures. ResNet and DenseNet can be considered as two special cases of our framework. Numerical experiments on CIFAR-10, CIFAR-100 and ImageNet verify the advantage of our optimization algorithm inspired structures over ResNet and DenseNet.

With the increasing of multi-modal data on the internet, cross-modal retrieval has received a lot of attention in recent years. It aims to use one type of data as query and retrieve results of another type. For different modality data, how to reduce their heterogeneous property and preserve their local relationship are two main challenges. In this paper, we present a novel joint dictionary learning and semantic constrained latent subspace learning method for cross-modal retrieval~(JDSLC) to deal with above two issues. In this unified framework, samples from different modalities are encoded by their corresponding dictionaries to reduce the semantic gap. In the meantime, we learn modality-specific projection matrices to map the sparse coefficients into the shared latent subspace. Meanwhile, we impose a novel cross-modal similarity constraint to make the representations of samples that belong to same class but from different modalities as close as possible in the latent subspace. An efficient algorithm is proposed to jointly optimize the proposed model and learn the optimal dictionary, coefficients and projection matrix for each modality. Extensive experimental results on multiple benchmark datasets show that our method outperforms the state-of-the-art approaches.

Improving information flow in deep networks helps to ease the training difficulties and utilize parameters more efficiently. Here we propose a new convolutional neural network architecture with alternately updated clique (CliqueNet). In contrast to prior networks, there are both forward and backward connections between any two layers in the same block. The layers are constructed as a loop and are updated alternately. The CliqueNet has some unique properties. For each layer, it is both the input and output of any other layer in the same block, so that the information flow among layers is maximized. During propagation, the newly updated layers are concatenated to re-update previously updated layer, and parameters are reused for multiple times. This recurrent feedback structure is able to bring higher level visual information back to refine low-level filters and achieve spatial attention. We analyze the features generated at different stages and observe that using refined features leads to a better result. We adopt a multiscale feature strategy that effectively avoids the progressive growth of parameters. Experiments on image recognition datasets including CIFAR-10, CIFAR-100, SVHN and ImageNet show that our proposed models achieve the state-of-the-art performance with fewer parameters.

The recent proposed Tensor Nuclear Norm (TNN) [Lu et al., 2016; 2018a] is an interesting convex penalty induced by the tensor SVD [Kilmer and Martin, 2011]. It plays a similar role as the matrix nuclear norm which is the convex surrogate of the matrix rank. Considering that the TNN based Tensor Robust PCA [Lu et al., 2018a] is an elegant extension of Robust PCA with a similar tight recovery bound, it is natural to solve other low rank tensor recovery problems extended from the matrix cases. However, the extensions and proofs are generally tedious. The general atomic norm provides a unified view of low-complexity structures induced norms, e.g., the l1-norm and nuclear norm. The sharp estimates of the required number of generic measurements for exact recovery based on the atomic norm are known in the literature. In this work, with a careful choice of the atomic set, we prove that TNN is a special atomic norm. Then by computing the Gaussian width of certain cone which is necessary for the sharp estimate, we achieve a simple bound for guaranteed low tubal rank tensor recovery from Gaussian measurements. Specifically, we show that by solving a TNN minimization problem, the underlying tensor of size n1×n2×n3 with tubal rank r can be exactly recovered when the given number of Gaussian measurements is O(r(n1+n2−r)n3). It is order optimal when comparing with the degrees of freedom r(n1+n2−r)n3. Beyond the Gaussian mapping, we also give the recovery guarantee of tensor completion based on the uniform random mapping by TNN minimization. Numerical experiments verify our theoretical results.

In this paper, we propose a new technique named \textit{Stochastic Path-Integrated Differential EstimatoR} (SPIDER), which can be used to track many deterministic quantities of interests with significantly reduced computational cost. Combining SPIDER with the method of normalized gradient descent, we propose SPIDER-SFO that solve non-convex stochastic optimization problems using stochastic gradients only. We provide a few error-bound results on its convergence rates. Specially, we prove that the SPIDER-SFO algorithm achieves a gradient computation cost of $\mathcal{O}\left( \min( n^{1/2} \epsilon^{-2}, \epsilon^{-3} ) \right)$ to find an $\epsilon$-approximate first-order stationary point. In addition, we prove that SPIDER-SFO nearly matches the algorithmic lower bound for finding stationary point under the gradient Lipschitz assumption in the finite-sum setting. Our SPIDER technique can be further applied to find an $(\epsilon, \mathcal{O}(\ep^{0.5}))$-approximate second-order stationary point at a gradient computation cost of $\tilde{\mathcal{O}}\left( \min( n^{1/2} \epsilon^{-2}+\epsilon^{-2.5}, \epsilon^{-3} ) \right)$.

Convolutional neural networks (CNNs) have recently achieved great success in single-image super-resolution (SISR). However, these methods tend to produce over-smoothed outputs and miss some textural details. To solve these problems, we propose the Super-Resolution CliqueNet (SRCliqueNet) to reconstruct the high resolution (HR) image with better textural details in the wavelet domain. The proposed SRCliqueNet firstly extracts a set of feature maps from the low resolution (LR) image by the clique blocks group. Then we send the set of feature maps to the clique up-sampling module to reconstruct the HR image. The clique up-sampling module consists of four sub-nets which predict the high resolution wavelet coefficients of four sub-bands. Since we consider the edge feature properties of four sub-bands, the four sub-nets are connected to the others so that they can learn the coefficients of four sub-bands jointly. Finally we apply inverse discrete wavelet transform (IDWT) to the output of four sub-nets at the end of the clique up-sampling module to increase the resolution and reconstruct the HR image. Extensive quantitative and qualitative experiments on benchmark datasets show that our method achieves superior performance over the state-of-the-art methods.

-norm based low rank matrix factorization in the presence of missing data and outliers remains a hot topic in computer vision. Due to non-convexity and non-smoothness, all the existing methods either lack scalability or robustness, or have no theoretical guarantee on convergence. In this paper, we apply the Majorization Minimization technique to solve this problem. At each iteration, we upper bound the original function with a strongly convex surrogate. By minimizing the surrogate and updating the iterates accordingly, the objective function has sufficient decrease, which is stronger than just being non-increasing that other methods could offer. As a consequence, without extra assumptions, we prove that any limit point of the iterates is a stationary point of the objective function. In comparison, other methods either do not have such a convergence guarantee or require extra critical assumptions. Extensive experiments on both synthetic and real data sets testify to the effectiveness of our algorithm. The speed of our method is also highly competitive.

Accompanied with the rising popularity of compressed sensing, the Alternating Direction Method of Multipliers (ADMM) has become the most widely used solver for linearly constrained convex problems with separable objectives. In this work, we observe that many existing ADMMs update the primal variable by minimizing different majorant functions with their convergence proofs given case by case. Inspired by the principle of majorization minimization, we respectively present the unified frameworks of Gauss-Seidel ADMMs and Jacobian ADMMs, which use different historical information for the current updating. Our frameworks generalize previous ADMMs to solve the problems with non-separable objectives. We also show that ADMMs converge faster when the used majorant function is tighter. We then propose the Mixed Gauss-Seidel and Jacobian ADMM (M-ADMM) which alleviates the slow convergence issue of Jacobian ADMMs by absorbing merits of the Gauss-Seidel ADMMs. M-ADMM can be further improved by backtracking and wise variable partition. We also propose to solve the multi-blocks problems by Proximal Gauss-Seidel ADMM which is of the Gauss-Seidel type. It convegences for non-strongly convex objective. Experiments on both synthesized and real-world data demonstrate the superiority of our new ADMMs. Finally, we release a toolbox that implements efficient ADMMs for many problems in compressed sensing.

The heavy-tailed distributions of corrupted outliers and singular values of all channels in low-level vision have proven effective priors for many applications such as background modeling, photometric stereo and image alignment. And they can be well modeled by a hyper-Laplacian. However, the use of such distributions generally leads to challenging non-convex, non-smooth and non-Lipschitz problems, and makes existing algorithms very slow for large-scale applications. Together with the analytic solutions to $\ell _{p}$ -norm minimization with two specific values of $p$ , i.e., $p=1/2$ and $p=2/3$ , we propose two novel bilinear factor matrix norm minimization models for robust principal component analysis. We first define the double nuclear norm and Frobenius/nuclear hybrid norm penalties, and then prove that they are in essence the Schatten- $1/2$ and $2/3$ quasi-norms, respectively, which lead to much more tractable and scalable Lipschitz optimization problems. Our experimental analysis shows that both our methods yield more accurate solutions than original Schatten quasi-norm minimization, even when the number of observations is very limited. Finally, we apply our penalties to various low-level vision problems, e.g., text removal, moving object detection, image alignment and inpainting, and show that our methods usually outperform the state-of-the-art methods.

Robust principal component analysis (RPCA) via decomposition into low-rank plus sparse matrices offers a powerful framework for a large variety of applications such as image processing, video processing, and 3-D computer vision. Indeed, most of the time these applications require to detect sparse outliers from the observed imagery data that can be approximated by a low-rank matrix. Moreover, most of the time experiments show that RPCA with additional spatial and/or temporal constraints often outperforms the state-of-the-art algorithms in these applications. Thus, the aim of this paper is to survey the applications of RPCA in computer vision. In the first part of this paper, we review representative image processing applications as follows: 1) low-level imaging such as image recovery and denoising, image composition, image colorization, image alignment and rectification, multifocus image, and face recognition; 2) medical imaging such as dynamic magnetic resonance imaging (MRI) for acceleration of data acquisition, background suppression, and learning of interframe motion fields; and 3) imaging for 3-D computer vision with additional depth information such as in structure from motion (SfM) and 3-D motion recovery. In the second part, we present the applications of RPCA in video processing which utilize additional spatial and temporal information compared to image processing. Specifically, we investigate video denoising and restoration, hyperspectral video, and background/foreground separation. Finally, we provide perspectives on possible future research directions and algorithmic frameworks that are suitable for these applications.

In practice, high-dimensional data are typically sampled from low-dimensional subspaces, but with intrusion of outliers and/or noises. Recovering the underlying structure and the pollution from the observations is of utmost importance to understanding the data. Besides properly modeling the subspace structure, how to handle the pollution is a core question regarding the recovery quality, the main origins of which include small dense noises and gross sparse outliers. Compared with the small noises, the outliers more likely ruin the recovery, as their arbitrary magnitudes can dominate the fidelity, and thus lead to misleading/erroneous results. Concerning the above, this paper concentrates on robust outlier estimate for low rank matrix recovery, termed as ROUTE. The principle is to classify each entry as an outlier or an inlier (with confidence). We formulate the outlier screening and the recovery into a unified framework. To seek the optimal solution to the problem, we first introduce a block coordinate descent based optimizer (ROUTE-BCD), then customize an alternating direction method of multipliers based one (ROUTE-ADMM). Through analyzing theoretical properties and practical behaviors, ROUTE-ADMM shows its superiority over ROUTE-BCD in terms of computational complexity, initialization insensitivity and recovery accuracy. Extensive experiments on both synthetic and real data are conducted to show the efficacy of our strategy and reveal its significant improvement over other state-of-the-art alternatives. Our code is publicly available at https://sites.google.com/view/xjguo/route.

Hashing is a useful technique for fast nearest neighbor search due to its low storage cost and fast query speed. Unsupervised hashing aims at learning binary hash codes for the original features so that the pairwise distances can be best preserved. While several works have targeted on this task, the results are not satisfactory mainly due to the over-simplified model. In this paper, we propose a unified and concise unsupervised hashing framework, called binary multidimensional scaling, which is able to learn the hash code for distance preservation in both batch and online mode. In the batch mode, unlike most existing hashing methods, we do not need to simplify the model by predefining the form of hash map. Instead, we learn the binary codes directly based on the pairwise distances among the normalized original features by alternating minimization. This enables a stronger expressive power of the hash map. In the online mode, we consider the holistic distance relationship between current query example and those we have already learned, rather than only focusing on current data chunk. It is useful when the data come in a streaming fashion. Empirical results show that while being efficient for training, our algorithm outperforms state-of-the-art methods by a large margin in terms of distance preservation, which is practical for real-world applications.

Sparse representation leads to an efficient way to approximately recover a signal by the linear composition of a few bases from a learnt dictionary based on which various successful applications have been achieved. However, in the scenario of data compression, its efficiency and popularity are hindered. It is because of the fact that encoding sparsely distributed coefficients may consume more bits for representing the index of nonzero coefficients. Therefore, introducing an accurate rate constraint in sparse coding and dictionary learning becomes meaningful, which has not been fully exploited in the context of sparse representation. According to the Shannon entropy inequality, the variance of Gaussian distributed data bound its entropy, indicating the actual bitrate can be well estimated by its variance. Hence, a globally variance-constrained sparse representation (GVCSR) model is proposed in this paper, where a variance-constrained rate term is introduced to the optimization process. Specifically, we employ the alternating direction method of multipliers (ADMMs) to solve the non-convex optimization problem for sparse coding and dictionary learning, both of them have shown the state-of-the-art rate-distortion performance for image representation. Furthermore, we investigate the potential of applying the GVCSR algorithm in the practical image set compression, where the optimized dictionary is trained to efficiently represent the images captured in similar scenarios by implicitly utilizing inter-image correlations. Experimental results have demonstrated superior rate-distortion performance against the state-of-the-art methods.

Recently, a tensor nuclear norm (TNN) based method was proposed to solve the tensor completion problem, which has achieved state-of-the-art performance on image and video inpainting tasks. However, it requires computing tensor singular value decomposition (t-SVD), which costs much computation and thus cannot efficiently handle tensor data, due to its natural large scale. Motivated by TNN, we propose a novel low-rank tensor factorization method for efficiently solving the 3-way tensor completion problem. Our method preserves the low-rank structure of a tensor by factorizing it into the product of two tensors of smaller sizes. In the optimization process, our method only needs to update two smaller tensors, which can be more efficiently conducted than computing t-SVD. Furthermore, we prove that the proposed alternating minimization algorithm can converge to a Karush-Kuhn-Tucker point. Experimental results on the synthetic data recovery, image and video inpainting tasks clearly demonstrate the superior performance and efficiency of our developed method over state-of-the-arts including the TNN and matricization methods.

Nowadays, asynchronous parallel algorithms have received much attention in the optimization field due to the crucial demands for modern large-scale optimization problems. However, most asynchronous algorithms focus on convex problems. Analysis on nonconvex problems is lacking. For the Asynchronous Stochastic Descent (ASGD) algorithm, the best result from (Lian et al., 2015) can only achieve an asymptotic O(\frac{1}{\epsilon^2}) rate (convergence to the stationary points) on nonconvex problems. In this paper, we study Stochastic Variance Reduced Gradient (SVRG) in the asynchronous setting. We propose the Asynchronous Stochastic Variance Reduced Gradient (ASVRG) algorithm for nonconvex finite-sum problems. We develop two schemes for ASVRG, depending on whether the parameters are updated as an atom or not. We prove that both of the two schemes can achieve linear speed up (a non-asymptotic O(\frac{n^\frac{2}{3}}{\epsilon}) rate to the stationary points) for nonconvex problems when the delay parameter \tau\leq n^{\frac{1}{3}}, where n is the number of training samples. We also establish a non-asymptotic O(\frac{n^\frac{2}{3}\tau^\frac{1}{3}}{\epsilon}) rate (convergence to the stationary points) for our algorithm without assumptions on \tau. This further demonstrates that even with asynchronous updating, SVRG has less number of Incremental First-order Oracles (IFOs) compared with Stochastic Gradient Descent and Gradient Descent. We also experiment on a shared memory multi-core system to demonstrate the efficiency of our algorithm.

The Schatten-p norm (0 < p < 1) has been widely used to replace the nuclear norm for better approximating the rank function. However, existing methods are either 1) not scalable for large scale problems due to relying on singular value decomposition (SVD) in every iteration, or 2) specific to some p values, e.g., 1/2, and 2/3. In this paper, we show that for any p, p1, and p2 > 0 satisfying 1/p = 1/p1 + 1/p2, there is an equivalence between the Schatten-p norm of one matrix and the Schatten-p1 and the Schatten-p2 norms of its two factor matrices. We further extend the equivalence to multiple factor matrices and show that all the factor norms can be convex and smooth for any p > 0. In contrast, the original Schatten-p norm for 0 < p < 1 is non-convex and non-smooth. As an example we conduct experiments on matrix completion. To utilize the convexity of the factor matrix norms, we adopt the accelerated proximal alternating linearized minimization algorithm and establish its sequence convergence. Experiments on both synthetic and real datasets exhibit its superior performance over the state-of-the-art methods. Its speed is also highly competitive.

We study the problem of recovering a t-sparse real vector from m quadratic equations yi=(ai*x)^2 with noisy measurements yi's. This is known as the problem of compressive phase retrieval, and has been widely applied to X-ray diffraction imaging, microscopy, quantum mechanics, etc. The challenge is to design a a) fast and b) noise-tolerant algorithms with c) near-optimal sample complexity. Prior work in this direction typically achieved one or two of these goals, but none of them enjoyed the three performance guarantees simultaneously. In this work, with a particular set of sensing vectors ai's, we give a provable algorithm that is robust to any bounded yet unstructured deterministic noise. Our algorithm requires roughly O(t) measurements and runs in O(tn*log (1/epsilon)) time, where epsilon is the error. This result advances the state-of-the-art work, and guarantees the applicability of our method to large datasets. Experiments on synthetic and real data verify our theory.

Sparse representation is efficient to approximately recover signals by a linear composition of a few bases from an over-complete dictionary. However, in the scenario of data compression, its efficiency and popularity are hindered due to the extra overhead for encoding the sparse coefficients. Therefore, how to establish an accurate rate model in sparse coding and dictionary learning becomes meaningful, which has been not fully exploited in the context of sparse representation. According to the Shannon entropy inequality, the variance of data source can bound its entropy, thus can reflect the actual coding bits. Therefore, a Globally Variance-Constrained Sparse Representation (GVCSR) model is proposed, where a variance-constrained rate term is introduced to the conventional sparse representation. To solve the non-convex optimization problem, we employ the Alternating Direction Method of Multipliers (ADMM) for sparse coding and dictionary learning, both of which have shown state-of-the-art rate-distortion performance in image representation.

In practice, even very high-dimensional data are typically sampled from low-dimensional subspaces but with intrusion of outliers and/or noises. Recovering the underlying structure and the pollution from the observations is key to understanding and processing such data. Besides properly modeling the low-rank structure of subspace, how to handle the pollution, is core regarding the performance of recovery. Often, the observed data is posed as a superimposition of the clean data and residual, while the residual can be roughly divided into two groups, including small dense noises and gross sparse outliers. Compared with small noises, outliers more likely ruin the recovery, as they can be arbitrarily large. By considering the above, this paper designs a method for recovering the low rank matrix with robust outlier estimation, termed as ROUTE, in a unified manner. Theoretical analysis on convergence and optimality, and experimental results on both synthetic and real data are provided to demonstrate the efficacy of our proposed method and show its superiority over other state-of-the-arts.

Structure from motion (SfM) is a crucial and widely studied problem in computer vision. Recently, the factorization framework for SfM was formulated as a low rank approximation problem: the rank of rescaled measurement matrix is always smaller than four. Since the rank function is non-convex, a common practice is to replace with its convex surrogate, i.e., the nuclear norm. However, nuclear norm sometimes gets unsatisfactory results. In this paper, we apply the recently proposed truncated nuclear norm to handle the factorization framework in a non-convex way, which heavily penalizes the singular values beyond the desired rank. We further introduce weighted ℓ1-norm to handle missing data and outliers uniformly. Based on truncated nuclear norm, we propose two factorization models for projective reconstruction and metric reconstruction, respectively. We also proposed an extremely efficient algorithm to tackle one of the optimization sub-problems. Extensive experiments on synthetic and real datasets verify the effectiveness of our method for projective and metric reconstructions. Our method achieves higher accuracy in 3D reconstruction and is more robust to missing data and outliers.

We study stochastic convex optimization subjected to linear equality constraints. Traditional Stochastic Alternating Direction Method of Multipliers and its Nesterov's acceleration scheme can only achieve ergodic O(1/\sqrt{K}) convergence rates, where K is the number of iteration. By introducing Variance Reduction (VR) techniques, the convergence rates improve to ergodic O(1/K). In this paper, we propose a new stochastic ADMM which elaborately integrates Nesterov's extrapolation and VR techniques. With Nesterov’s extrapolation, our algorithm can achieve a non-ergodic O(1/K) convergence rate which is optimal for separable linearly constrained non-smooth convex problems, while the convergence rates of VR based ADMM methods are actually tight O(1/\sqrt{K}) in non-ergodic sense. To the best of our knowledge, this is the first work that achieves a truly accelerated, stochastic convergence rate for constrained convex problems. The experimental results demonstrate that our algorithm is significantly faster than the existing state-of-the-art stochastic ADMM methods.

Cross-modal retrieval has received much attention in recent years. It is a commonly used method to project multi-modality data into a common subspace and then retrieve. However, nearly all existing methods directly adopt the space defined by the binary class label information without learning as the shared subspace for regression. In this paper, we first adopt the spectral regression method to learn the optimal latent space shared by data of all modalities based on the orthogonal constraints. Then we construct a graph model to project the multi-modality data into the latent space. Finally, we combine these two processes together to jointly learn the latent space and regress. We conduct extensive experiments on multiple benchmark datasets and our proposed method outperforms the state-of-the-art approaches.

In most of existing digital cameras, color images have to be reconstructed from raw images which only have one color sensed at each pixel, as their imaging sensors are covered by color filter arrays (CFAs). At each pixel a CFA usually allows only a portion of the light spectrum to pass through and thereby reduces the light sensitivity of pixels. To address this issue, previous works have explored adding panchromatic pixels into CFAs. However, almost all existing methods assign panchromatic pixels empirically, making the designed CFAs prone to aliasing artifacts. In this paper, based on a mathematical model we propose a fully automatic approach to designing high-sensitivity CFAs using panchromatic pixels. By the frequency structure representation of CFAs, we formulate high-sensitivity CFA design as a continuous multi-objective optimization problem, where robustness to aliasing artifacts and percentage of panchromatic pixels are simultaneously maximized. We analyze the characteristics of our new formulation. According to the analysis, we develop a new method to propose frequency structure candidates, which can produce CFAs that reach a desired percentage of panchromatic pixels. Then for each candidate, we optimize parameters to obtain the final CFA, which is an appropriately balanced solution to the multi-objective optimization problem. We formulate the two design procedures as constrained optimization problems and solve them using the alternating direction method (ADM). Extensive experiments confirm the advantage of the proposed method in both low-light and normal-light conditions.

Demosaicking is the problem of reconstructing a color image from the raw image captured by a digital color camera that covers its only imaging sensor with a color filter array (CFA). Sparse representation-based demosaicking has been shown to produce superior reconstruction quality. However, almost all existing algorithms in this category use the CFAs, which are not specifically optimized for the algorithms. In this paper, we consider optimally designing CFAs for sparse representation-based demosaicking, where the dictionary is well-chosen. The fact that CFAs correspond to the projection matrices used in compressed sensing inspires us to optimize CFAs via minimizing the mutual coherence. This is more challenging than that for traditional projection matrices because CFAs have physical realizability constraints. However, most of the existing methods for minimizing the mutual coherence require that the projection matrices should be unconstrained, making them inapplicable for designing CFAs. We consider directly minimizing the mutual coherence with the CFA's physical realizability constraints as a generalized fractional programming problem, which needs to find sufficiently accurate solutions to a sequence of nonconvex nonsmooth minimization problems. We adapt the redistributed proximal bundle method to address this issue. Experiments on benchmark images testify to the superiority of the proposed method. In particular, we show that a simple sparse representation-based demosaicking algorithm with our specifically optimized CFA can outperform LSSC [1]. To the best of our knowledge, it is the first sparse representation-based demosaicking algorithm that beats LSSC in terms of CPSNR.

Mismatch removal is a key step in many computer vision problems. In this paper, we handle the mismatch removal problem by adopting shape interaction matrix (SIM). Given the homogeneous coordinates of the two corresponding point sets, we first compute the SIMs of the two point sets. Then, we detect the mismatches by picking out the most different entries between the two SIMs. Even under strong affine transformations, outliers, noises, and burstiness, our method can still work well. Actually, this paper is the first non-iterative mismatch removal method that achieves affine invariance. Extensive results on synthetic 2D points matching data sets and real image matching data sets verify the effectiveness, efficiency, and robustness of our method in removing mismatches. Moreover, when applied to partial-duplicate image search, our method reaches higher retrieval precisions with shorter time cost compared with the state-of-the-art geometric verification methods.

Most supervised dictionary learning methods optimize the combinations of reconstruction error, sparsity prior, and discriminative terms. Thus, the learnt dictionaries may not be optimal for recognition tasks. Also, the sparse codes learning models in the training and the testing phases are inconsistent. Besides, without utilizing the intrinsic data structure, many dictionary learning methods only employ the$\ell _{0}$or$\ell _{1}$norm to encode each datum independently, limiting the performance of the learnt dictionaries. We present a novel bilevel model-based discriminative dictionary learning method for recognition tasks. The upper level directly minimizes the classification error, while the lower level uses the sparsity term and the Laplacian term to characterize the intrinsic data structure. The lower level is subordinate to the upper level. Therefore, our model achieves an overall optimality for recognition in that the learnt dictionary is directly tailored for recognition. Moreover, the sparse codes learning models in the training and the testing phases can be the same. We further propose a novel method to solve our bilevel optimization problem. It first replaces the lower level with its Karush-Kuhn–Tucker conditions and then applies the alternating direction method of multipliers to solve the equivalent problem. Extensive experiments demonstrate the effectiveness and robustness of our method.

In the literature, most existing graph-based semi-supervised learning methods only use the label information of observed samples in the label propagation stage, while ignoring such valuable information when learning the graph. In this paper, we argue that it is beneficial to consider the label information in the graph learning stage. Specifically, by enforcing the weight of edges between labeled samples of different classes to be zero, we explicitly incorporate the label information into the state-of-the-art graph learning methods, such as the low-rank representation (LRR), and propose a novel semi-supervised graph learning method called semi-supervised low-rank representation. This results in a convex optimization problem with linear constraints, which can be solved by the linearized alternating direction method. Though we take LRR as an example, our proposed method is in fact very general and can be applied to any self-representation graph learning methods. Experiment results on both synthetic and real data sets demonstrate that the proposed graph learning method can better capture the global geometric structure of the data, and therefore is more effective for semi-supervised learning tasks.

The Augmented Lagragian Method (ALM) and Alternating Direction Method of Multiplier (ADMM) have been powerful optimization methods for general convex programming subject to linear constraint. We consider the convex problem whose objective consists of a smooth part and a nonsmooth but simple part. We propose the Fast Proximal Augmented Lagragian Method (Fast PALM) which achieves the convergence rate O(1/K2), compared with O(1/K) by the traditional PALM. In order to further reduce the per-iteration complexity and handle the multi-blocks problem, we propose the Fast Proximal ADMM with Parallel Splitting (Fast PL-ADMM-PS) method. It also partially improves the rate related to the smooth part of the objective function. Experimental results on both synthesized and real world data demonstrate that our fast methods significantly improve the previous PALM and ADMM

We propose a new majorization-minimization (MM) method for non-smooth and non-convex programs, which is general enough to include the existing MM methods. Besides the local majorization condition, we only require that the difference between the directional derivatives of the objective function and its surrogate function vanishes when the number of iterations approaches infinity, which is a very weak condition. So our method can use a surrogate function that directly approximates the non-smooth objective function. In comparison, all the existing MM methods construct the surrogate function by approximating the smooth component of the objective function. We apply our relaxed MM methods to the robust matrix factorization (RMF) problem with different regularizations, where our locally majorant algorithm shows advantages over the state-of-the-art approaches for RMF. This is the first algorithm for RMF ensuring, without extra assumptions, that any limit point of the iterates is a stationary point.

This paper studies the Tensor Robust Principal Component (TRPCA) problem which extends the known Robust PCA [4] to the tensor case. Our model is based on a new tensor Singular Value Decomposition (t-SVD) [14] and its induced tensor tubal rank and tensor nuclear norm. Consider that we have a 3-way tensor X ε Rn1×n2×n3such that X = L0+ S0, where L0has low tubal rank andS0 issparse. Is that possible to recover both components? In this work, we prove that under certain suitable assumptions, we can recover both the low-rank and the sparse components exactly by simply solving a convex program whose objective is a weighted combination of the tensor nuclear norm and the l1-norm, i.e., min L, E ||L||* + λ||ε||1, s.t. X = L + ε, where λ = 1/√max(n1, n2)n3. Interestingly, TRPCA involves RPCA as a special case when n3= 1 and thus it is a simple and elegant tensor extension of RPCA. Also numerical experiments verify our theory and the application for the image denoising demonstrates the effectiveness of our method.

Estimating blur kernels from real world images is a challenging problem as the linear image formation assumption does not hold when significant outliers, such as saturated pixels and non-Gaussian noise, are present. While some existing non-blind deblurring algorithms can deal with outliers to a certain extent, few blind deblurring methods are developed to well estimate the blur kernels from the blurred images with outliers. In this paper, we present an algorithm to address this problem by exploiting reliable edges and removing outliers in the intermediate latent images, thereby estimating blur kernels robustly. We analyze the effects of outliers on kernel estimation and show that most state-of-the-art blind deblurring methods may recover delta kernels when blurred images contain significant outliers. We propose a robust energy function which describes the properties of outliers for the final latent image restoration. Furthermore, we show that the proposed algorithm can be applied to improve existing methods to deblur images with outliers. Extensive experiments on different kinds of challenging blurry images with significant amount of outliers demonstrate the proposed algorithm performs favorably against the state-of-the-art methods.

It is a very challenging task to recognize emotion in the wild. Recently, combining information from various views or modalities has attracted more attention. Cross modality features and features extracted by different methods are regarded as multi-view information of the sample. In this paper, we propose a method to analyse multi-view features of emotion samples and automatically recognize the expression as part of the fourth Emotion Recognition in the Wild Challenge (EmotiW 2016). In our method, we first extract multi-view features such as BoF, CNN, LBP-TOP and audio features for each expression sample. Then we learn the corresponding projection matrices to map multi-view features into a common subspace. In the meantime, we impose l_2,1-norm penalties on projection matrices for feature selection. We apply both this method and PLSR to emotion recognition. We conduct experiments on both AFEW and HAPPEI datasets, and achieve superior performance. The best recognition accuracy of our method is 0.5531 on the AFEW dataset for video based emotion recognition in the wild. The minimum RMSE for group happiness intensity recognition is 0.9525 on HAPPEI dataset. Both of them are much better than that of the challenge baseline.

Annotating images with tags is useful for indexing and retrieving images. However, many available annotation data include missing or inaccurate annotations. In this paper, we propose an image annotation framework which sequentially performs tag completion and refinement. We utilize the subspace property of data via sparse subspace clustering for tag completion. Then we propose a novel matrix completion model for tag refinement, integrating visual correlation, semantic correlation and the novelly studied property of complex errors. The proposed method outperforms the state-of-the-art approaches on multiple benchmark datasets even when they contain certain levels of annotation noise.

Many problems in computer vision can be formulated as multidimensional ellipsoid-specific fitting, which is to minimize the residual error such that the underlying quadratic surface is a multidimensional ellipsoid. In this paper, we present a fast and robust algorithm for solving ellipsoid-specific fitting directly. Our method is based on the alternating direction method of multipliers, which does not introduce extra positive semi-definiteness constraints. The computation complexity is thus significantly lower than those of semi-definite programming (SDP) based methods. More specifically, to fit n data points into a p dimensional ellipsoid, our complexity is O(p(6) + np(4))+O(p(3)), where the former O results from preprocessing data once, while that of the state-of-the-art SDP method is O(p(6) + np(4) + n(3/2)p(2)) for each iteration. The storage complexity of our algorithm is about 1/2np(2), which is at most 1/4 of those of SDP methods. Extensive experiments testify to the great speed and accuracy advantages of our method over the state-of-the-art approaches. The implementation of our method is also much simpler than SDP based methods.

Partial differential equations (PDEs) have been used to formulate image processing for several decades. Generally, a PDE system consists of two components: the governing equation and the boundary condition. In most previous work, both of them are generally designed by people using mathematical skills. However, in real world visual analysis tasks, such predefined and fixed-form PDEs may not be able to describe the complex structure of the visual data. More importantly, it is hard to incorporate the labeling information and the discriminative distribution priors into these PDEs. To address above issues, we propose a new PDE framework, named learning to diffuse (LTD), to adaptively design the governing equation and the boundary condition of a diffusion PDE system for various vision tasks on different types of visual data. To our best knowledge, the problems considered in this paper (i.e., saliency detection and object tracking) have never been addressed by PDE models before. Experimental results on various challenging benchmark databases show the superiority of LTD against existing state-of-the-art methods for all the tested visual analysis tasks.

Low-rank representation (LRR) has recently attracted a great deal of attention due to its pleasing efficacy in exploring low-dimensional subspace structures embedded in data. For a given set of observed data corrupted with sparse errors, LRR aims at learning a lowest-rank representation of all data jointly. LRR has broad applications in pattern recognition, computer vision and signal processing. In the real world, data often reside on low-dimensional manifolds embedded in a high-dimensional ambient space. However, the LRR method does not take into account the non-linear geometric structures within data, thus the locality and similarity information among data may be missing in the learning process. To improve LRR in this regard, we propose a general Laplacian regularized low-rank representation framework for data representation where a hypergraph Laplacian regularizer can be readily introduced into, i.e., a Non-negative Sparse Hyper-Laplacian regularized LRR model (NSHLRR). By taking advantage of the graph regularizer, our proposed method not only can represent the global low-dimensional structures, but also capture the intrinsic non-linear geometric information in data. The extensive experimental results on image clustering, semi-supervised image classification and dimensionality reduction tasks demonstrate the effectiveness of the proposed method.

In digital color imaging, the raw image is typically obtained through a single sensor covered by a color filter array (CFA), which allows only one color component to be measured at each pixel. The procedure to reconstruct a full color image from the raw image is known as demosaicking. Since the CFA may cause irreversible visual artifacts, the CFA and the demosaicking algorithm are crucial to the quality of demosaicked images. Fortunately, the design of CFAs in the frequency domain provides a theoretical approach to handling this issue. However, almost all the existing design methods in the frequency domain involve considerable human effort. In this paper, we present a new method to automatically design CFAs in the frequency domain. Our method is based on the frequency structure representation of mosaicked images. We utilize a multi-objective optimization approach to propose frequency structure candidates, in which the overlap among the frequency components of images mosaicked with the CFA is minimized. Then, we optimize parameters for each candidate, which is formulated as a constrained optimization problem. We use the alternating direction method to solve it. Our parameter optimization method is applicable to arbitrary frequency structures, including those with conjugate replicas of chrominance components. Experiments on benchmark images confirm the advantage of the proposed method.

The nuclear norm is widely used as a convex surrogate of the rank function in compressive sensing for low rank matrix recovery with its applications in image recovery and signal processing. However, solving the nuclear norm-based relaxed convex problem usually leads to a suboptimal solution of the original rank minimization problem. In this paper, we propose to use a family of nonconvex surrogates of L0-norm on the singular values of a matrix to approximate the rank function. This leads to a nonconvex nonsmooth minimization problem. Then, we propose to solve the problem by an iteratively re-weighted nuclear norm (IRNN) algorithm. IRNN iteratively solves a weighted singular value thresholding problem, which has a closed form solution due to the special properties of the nonconvex surrogate functions. We also extend IRNN to solve the nonconvex problem with two or more blocks of variables. In theory, we prove that the IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthesized data and real images demonstrate that IRNN enhances the low rank matrix recovery compared with the state-of-the-art convex algorithms.

Spectral clustering (SC) is one of the most widely used methods for data clustering. It first finds a low-dimensional embedding U of data by computing the eigenvectors of the normalized Laplacian matrix, and then performs k-means on UTto get the final clustering result. In this paper, we observe that, in the ideal case, UUTshould be block diagonal and thus sparse. Therefore, we propose the sparse SC (SSC) method that extends the SC with sparse regularization on UUT. To address the computational issue of the nonconvex SSC model, we propose a novel convex relaxation of SSC based on the convex hull of the fixed rank projection matrices. Then, the convex SSC model can be efficiently solved by the alternating direction method of multipliers Furthermore, we propose the pairwise SSC that extends SSC to boost the clustering performance by using the multi-view information of data. Experimental comparisons with several baselines on real-world datasets testify to the efficacy of our proposed methods.

Subspace recovery from the corrupted and missing data is crucial for various applications in signal processing and information theory. To complete missing values and detect column corruptions, the existing robust matrix completion (MC) methods mostly concentrate on recovering a low-rank matrix from a few corrupted coefficients with respect to standard basis, which, however, does not apply to more general basis, e.g., Fourier basis. In this paper, we prove that the range space of an m × n matrix with rank r can be exactly recovered from a few coefficients with respect to general basis, though r and the number of corrupted samples are both as high as O(min{m, n}/ log3(m + n)). Our model covers the previous ones as special cases, and robust MC can recover the intrinsic matrix with a higher rank. Moreover, we suggest a universal choice of the regularization parameter, which is λ = 1/√log n. By our ℓ2,1filtering algorithm, which has theoretical guarantees, we can further reduce the computational cost of our model. As an application, we also find that the solutions to extended robust low-rank representation and to our extended robust MC are mutually expressible, so both our theory and algorithm can be applied to the subspace clustering problem with missing values under certain conditions. The experiments verify our theories.

Subspace clustering groups a set of samples from a union of several linear subspaces into clusters, so that the samples in the same cluster are drawn from the same linear subspace. In the majority of the existing work on subspace clustering, clusters are built based on feature information, while sample correlations in their original spatial structure are simply ignored. Besides, original high-dimensional feature vector contains noisy/redundant information, and the time complexity grows exponentially with the number of dimensions. To address these issues, we propose a tensor low-rank representation (TLRR) and sparse coding-based (TLRRSC) subspace clustering method by simultaneously considering feature information and spatial structures. TLRR seeks the lowest rank representation over original spatial structures along all spatial directions. Sparse coding learns a dictionary along feature spaces, so that each sample can be represented by a few atoms of the learned dictionary. The affinity matrix used for spectral clustering is built from the joint similarities in both spatial and feature spaces. TLRRSC can well capture the global structure and inherent feature information of data and provide a robust subspace segmentation from corrupted data. Experimental results on both synthetic and real-world data sets show that TLRRSC outperforms several established stateof- the-art methods.

Feature learning plays a central role in pattern recognition. In recent years, many representation-based feature learning methods have been proposed and have achieved great success in many applications. However, these methods perform feature learning and subsequent classification in two separate steps, which may not be optimal for recognition tasks. In this paper, we present a supervised low-rank-based approach for learning discriminative features. By integrating latent low-rank representation (LatLRR) with a ridge regression-based classifier, our approach combines feature learning with classification, so that the regulated classification error is minimized. In this way, the extracted features are more discriminative for the recognition tasks. Our approach benefits from a recent discovery on the closed-form solutions to noiseless LatLRR. When there is noise, a robust Principal Component Analysis (PCA)-based denoising step can be added as preprocessing. When the scale of a problem is large, we utilize a fast randomized algorithm to speed up the computation of robust PCA. Extensive experimental results demonstrate the effectiveness and robustness of our method.

This work studies the Generalized Singular Value Thresholding (GSVT) operator associated with a nonconvex function g defined on the singular values of X. We prove that GSVT can be obtained by performing the proximal operator of g on the singular values since Proxg(.) is monotone when g is lower bounded. If the nonconvex g satisfies some conditions (many popular nonconvex surrogate functions, e.g., lp-norm, 0 < p < 1, of l0-norm are special cases), a general solver to find Proxg(b) is proposed for any b ≥ 0. GSVT greatly generalizes the known Singular Value Thresholding (SVT) which is a basic subroutine in many convex low rank minimization methods. We are able to solve the nonconvex low rank minimization problem by using GSVT in place of SVT.

Subspace recovery from noisy or even corrupted data is critical for various applications in machine learning and data analysis. To detect outliers, Robust PCA (R PCA) via Outlier Pursuit was proposed and had found many successful applications. However, the current theoretical analysis on Outlier Pursuit only shows that it succeeds when the sparsity of the corruption matrix is of O(n/r), where n is the number of the samples and r is the rank of the intrinsic matrix which may be comparable to n. Moreover, the regularization parameter is suggested as 3/(7 squareroot gamma n}, where gamma is a parameter that is not known a priori. In this paper, with incoherence condition and proposed ambiguity condition we prove that Outlier Pursuit succeeds when the rank of the intrinsic matrix is of O(n log n) and the sparsity of the corruption matrix is of O(n). We further show that the orders of both bounds are tight. Thus R-PCA via Outlier Pursuit is able to recover intrinsic matrix of higher rank and identify much denser corruptions than what the existing results could predict. Moreover, we suggest that the regularization parameter be chosen as 1 squareroot{log n}, which is definite. Our analysis waives the necessity of tuning the regularization parameter and also significantly extends the working range of the Outlier Pursuit. Experiments on synthetic and real data verify our theories.

The Riemannian three-factor matrix completion (R3MC) algorithm is one of the state-of-the-art geometric optimization methods for the low-rank matrix completion problem. It is a nonlinear conjugate-gradient method optimizing on a quotient Riemannian manifold. In the line search step, R3MC approximates the minimum point on the searching curve by minimizing on the line tangent to the curve. However, finding the exact minimum point by iteration is too expensive. We address this issue by proposing a new retraction with a minimizing property. This special property provides the exact minimization for the line search by establishing correspondences between points on the searching curve and points on the tangent line. Accelerated R3MC, which is R3MC equipped with this new retraction, outperforms the original algorithm and other geometric algorithms for matrix completion in our empirical study.

Subspace clustering is a problem of finding a multi-subspace representation that best fits sample points drawn from a high-dimensional space. The existing clustering models generally adopt different norms to describe noise, which is equivalent to assuming that the data are corrupted by specific types of noise. In practice, however, noise is much more complex. So it is inappropriate to simply use a certain norm to model noise. Therefore, we propose Mixture of Gaussian Regression (MoG Regression) for subspace clustering by modeling noise as a Mixture of Gaussians (MoG). The MoG Regression provides an effective way to model a much broader range of noise distributions. As a result, the obtained affinity matrix is better at characterizing the structure of data in real applications. Experimental results on multiple datasets demonstrate that MoG Regression significantly outperforms state-of-the-art subspace clustering methods.

State of the art approaches for Semi-Supervised Learning (SSL) usually follow a two-stage framework -- constructing an affinity matrix from the data and then propagating the partial labels on this affinity matrix to infer those unknown labels. While such a two-stage framework has been successful in many applications, solving two subproblems separately only once is still suboptimal because it does not fully exploit the correlation between the affinity and the labels. In this paper, we formulate the two stages of SSL into a unified optimization framework, which learns both the affinity matrix and the unknown labels simultaneously. In the unified framework, both the given labels and the estimated labels are used to learn the affinity matrix and to infer the unknown labels. We solve the unified optimization problem via an alternating direction method of multipliers combined with label propagation. Extensive experiments on a synthetic data set and several benchmark data sets demonstrate the effectiveness of our approach.

Emotion recognition in the wild is a very challenging task. In this paper, we propose a multiple models fusion method to automatically recognize the expression in the video clip as part of the third Emotion Recognition in the Wild Challenge (EmotiW 2015). In our method, we first extract dense SIFT, LBP-TOP and audio features from each video clip. For dense SIFT features, we use the bag of features (BoF) model with two different encoding methods (locality-constrained linear coding and group saliency based coding) to further represent it. During the classification process, we use partial least square regression to calculate the regression value of each model. By learning the optimal weight of each model based on the regression value, we fuse these models together. We conduct experiments on the given validation and test datasets, and achieve superior performance. The best recognition accuracy of our fusion method is 52.50% on the test dataset, which is 13.17% higher than the challenge baseline accuracy of 39.33%.

Optimization on Riemannian manifolds is an intuitive generalization of the traditional optimization algorithms in Euclidean spaces. In these algorithms, minimizing along a search direction becomes minimizing along a search curve lying on a manifold. Computing such a curve to be subsequently searched upon is itself computational intensive. We propose a new minimization scheme aiming to find a better step size utilizing the first order information of the search curve. We prove that this scheme can provide further reduction for the cost function when the retraction and the vector transport are collinear. Then we adapt this scheme to propose a heuristic strategy for line search. In numerical experiments, we apply this heuristic strategy to one of the geometric algorithms for matrix completion and show its feasibility and the potential in accelerating computation.

Nonconvex and nonsmooth problems have recently received considerable attention in signal/image processing, statistics and machine learning. However, solving the nonconvex and nonsmooth optimization problems remains a big challenge. Accelerated proximal gradient (APG) is an excellent method for convex programming. However, it is still unknown whether the usual APG can ensure the convergence to a critical point in nonconvex programming. To address this issue, we introduce a monitor-corrector step and extend APG for general nonconvex and nonsmooth programs. Accordingly, we propose a monotone APG and a non-monotone APG. The latter waives the requirement on monotonic reduction of the objective function and needs less computation in each iteration. To the best of our knowledge, we are the first to provide APG-type algorithms for general nonconvex and nonsmooth problems ensuring that every accumulation point is a critical point, and the convergence rates remain $O(1/k^2)$ when the problems are convex, in which k is the number of iterations. Numerical results testify to the advantage of our algorithms in speed.

Tag-based image retrieval (TBIR) has drawn much attention in recent years due to the explosive amount of digital images and crowdsourcing tags. However, the TBIR applications still suffer from the deficient and inaccurate tags provided by users. Inspired by the subspace clustering methods, we formulate the tag completion problem in a subspace clustering model which assumes that images are sampled from subspaces, and complete the tags using the state-of-the-art Low Rank Representation (LRR) method. And we propose a matrix completion algorithm to further refine the tags. Our empirical results on multiple benchmark datasets for image annotation show that the proposed algorithm outperforms state-of-the-art approaches when handling missing and noisy tags.

Recovering intrinsic low-dimensional subspaces from data distributed on them is a key preprocessing step to many applications. In recent years, a lot of work has modeled subspace recovery as low-rank minimization problems. We find that some representative models, such as robust principal component analysis (R-PCA), robust low-rank representation (R-LRR), and robust latent low-rank representation (R-LatLRR), are actually deeply connected. More specifically, we discover that once a solution to one of the models is obtained, we can obtain the solutions to other models in closed-form formulations. Since R-PCA is the simplest, our discovery makes it the center of low-rank subspace recovery models. Our work has two important implications. First, R-PCA has a solid theoretical foundation. Under certain conditions, we could find globally optimal solutions to these low-rank models at an overwhelming probability, although these models are nonconvex. Second, we can obtain significantly faster algorithms for these models by solving R-PCA first. The computation cost can be further cut by applying low-complexity randomized algorithms, for example, our novel l2,1 filtering algorithm, to R-PCA. Although for the moment the formal proof of our l2,1 filtering algorithm is not yet available, experiments verify the advantages of our algorithm over other state-of-the-art methods based on the alternating direction method.

The Penrose pixel layout, an aperiodic pixel layout in rhombus Penrose tiling, has been shown to substantially outperform the existing square pixel layout in super-resolution. However, it was tested only on grayscale images. To study its performance on color images, we have to reconstruct regular color images from Penrose raw images, i.e., images with only one color component at each Penrose pixel, resulting in the problem of demosaicking from Penrose pixels. Penrose demosaicking is more difficult than regular demosaicking, because none of the color components of the reconstructed regular color images are available. Therefore, most of the traditional demosaicking methods do not apply. We develop a sparse representation-based method for Penrose demosaicking. Extensive experiments show that Penrose pixel layout outperforms regular pixel layouts in terms of both perceptual evaluation and S-CIELAB. The Penrose pixel layout is unique among all irregular layouts because it is uniformly three-colorable and it has only two pixel shapes, thick and thin rhombi, making its manufacturing relatively easy.

This paper presents a general framework for solving the low-rank and/or sparse matrix minimization problems, which may involve multiple nonsmooth terms. The iteratively reweighted least squares (IRLSs) method is a fast solver, which smooths the objective function and minimizes it by alternately updating the variables and their weights. However, the traditional IRLS can only solve a sparse only or low rank only minimization problem with squared loss or an affine constraint. This paper generalizes IRLS to solve joint/mixed low-rank and sparse minimization problems, which are essential formulations for many tasks. As a concrete example, we solve the Schatten-p norm and l2,q-norm regularized low-rank representation problem by IRLS, and theoretically prove that the derived solution is a stationary point (globally optimal if p,q ≥ 1). Our convergence proof of IRLS is more general than previous one that depends on the special properties of the Schatten-p norm and l2,q-norm. Extensive experiments on both synthetic and real data sets demonstrate that our IRLS is much more efficient.

The sparse coding technique has shown flexibility and capability in image representation and analysis. It is a powerful tool in many visual applications. Some recent work has shown that incorporating the properties of task (such as discrimination for classification task) into dictionary learning is effective for improving the accuracy. However, the traditional supervised dictionary learning methods suffer from high computation complexity when dealing with large number of categories, making them less satisfactory in large scale applications. In this paper, we propose a novel multi-level discriminative dictionary learning method and apply it to large scale image classification. Our method takes advantage of hierarchical category correlation to encode multi-level discriminative information. Each internal node of the category hierarchy is associated with a discriminative dictionary and a classification model. The dictionaries at different layers are learnt to capture the information of different scales. Moreover, each node at lower layers also inherits the dictionary of its parent, so that the categories at lower layers can be described with multi-scale information. The learning of dictionaries and associated classification models is jointly conducted by minimizing an overall tree loss. The experimental results on challenging data sets demonstrate that our approach achieves excellent accuracy and competitive computation cost compared with other sparse coding methods for large scale image classification.

Low-rank representation (LRR) has received considerable attention in subspace segmentation due to its effectiveness in exploring low-dimensional subspace structures embedded in data. To preserve the intrinsic geometrical structure of data, a graph regularizer has been introduced into LRR framework for learning the locality and similarity information within data. However, it is often the case that not only the high-dimensional data reside on a non-linear low-dimensional manifold in the ambient space, but also their features lie on a manifold in feature space. In this paper, we propose a dual graph regularized LRR model (DGLRR) by enforcing preservation of geometric information in both the ambient space and the feature space. The proposed method aims for simultaneously considering the geometric structures of the data manifold and the feature manifold. Furthermore, we extend the DGLRR model to include non-negative constraint, leading to a parts-based representation of data. Experiments are conducted on several image data sets to demonstrate that the proposed method outperforms the state-of-the-art approaches in image clustering.

This paper aims at constructing a good graph to discover the intrinsic data structures under a semisupervised learning setting. First, we propose to build a nonnegative low-rank and sparse (referred to as NNLRS) graph for the given data representation. In particular, the weights of edges in the graph are obtained by seeking a nonnegative low-rank and sparse reconstruction coefficients matrix that represents each data sample as a linear combination of others. The so-obtained NNLRS-graph captures both the global mixture of subspaces structure (by the low-rankness) and the locally linear structure (by the sparseness) of the data, hence it is both generative and discriminative. Second, as good features are extremely important for constructing a good graph, we propose to learn the data embedding matrix and construct the graph simultaneously within one framework, which is termed as NNLRS with embedded features (referred to as NNLRS-EF). Extensive NNLRS experiments on three publicly available data sets demonstrate that the proposed method outperforms the state-of-the-art graph construction method by a large margin for both semisupervised classification and discriminative analysis, which verifies the effectiveness of our proposed method.

This paper proposes the Proximal Iteratively REweighted (PIRE) algorithm for solving a general problem, which involves a large body of nonconvex sparse and structured sparse related problems. Comparing with previous iterative solvers for nonconvex sparse problem, PIRE is much more general and efficient. The computational cost of PIRE in each iteration is usually as low as the state-of-the-art convex solvers. We further propose the PIRE algorithm with Parallel Splitting (PIRE-PS) and PIRE algorithm with Alternative Updating (PIRE-AU) to handle the multi-variable problems. In theory, we prove that our proposed methods converge and any limit solution is a stationary point. Extensive experiments on both synthesis and real data sets demonstrate that our methods achieve comparative learning performance, but are much more efficient, by comparing with previous nonconvex solvers.

The subspace segmentation problem is addressed in this paper by effectively constructing an exactly block-diagonal sample affinity matrix. The block-diagonal structure is heavily desired for accurate sample clustering but is rather difficult to obtain. Most current state-of-the-art subspace segmentation methods (such as SSC[4] and LRR[12]) resort to alternative structural priors (such as sparseness and low-rankness) to construct the affinity matrix. In this work, we directly pursue the block-diagonal structure by proposing a graph Laplacian constraint based formulation, and then develop an efficient stochastic subgradient algorithm for optimization. Moreover, two new subspace segmentation methods, the block-diagonal SSC and LRR, are devised in this work. To the best of our knowledge, this is the first research attempt to explicitly pursue such a block-diagonal structure. Extensive experiments on face clustering, motion segmentation and graph construction for semi-supervised learning clearly demonstrate the superiority of our novelly proposed subspace segmentation methods.

Subspace clustering is a powerful technology for clustering data according to the underlying subspaces. Representation based methods are the most popular subspace clustering approach in recent years. In this paper, we analyze the grouping effect of representation based methods in depth. In particular, we introduce the enforced grouping effect conditions, which greatly facilitate the analysis of grouping effect. We further find that grouping effect is important for subspace clustering, which should be explicitly enforced in the data self-representation model, rather than implicitly implied by the model as in some prior work. Based on our analysis, we propose the SMooth Representation (SMR) model. We also propose a new affinity measure based on the grouping effect, which proves to be much more effective than the commonly used one. As a result, our SMR significantly outperforms the state-of-the-art ones on benchmark datasets.

Partial Differential Equations (PDEs) have been successful in solving many low-level vision tasks. However, it is a challenging task to directly utilize PDEs for visual saliency detection due to the difficulty in incorporating human perception and high-level priors to a PDE system. Instead of designing PDEs with fixed formulation and boundary condition, this paper proposes a novel framework for adaptively learning a PDE system from an image for visual saliency detection. We assume that the saliency of image elements can be carried out from the relevances to the saliency seeds (i.e., the most representative salient elements). In this view, a general Linear Elliptic System with Dirichlet boundary (LESD) is introduced to model the diffusion from seeds to other relevant points. For a given image, we first learn a guidance map to fuse human prior knowledge to the diffusion system. Then by optimizing a discrete submodular function constrained with this LESD and a uniform matroid, the saliency seeds (i.e., boundary conditions) can be learnt for this image, thus achieving an optimal PDE system to model the evolution of visual saliency. Experimental results on various challenging image sets show the superiority of our proposed learning-based PDEs for visual saliency detection.

As surrogate functions of L0-norm, many nonconvex penalty functions have been proposed to enhance the sparse vector recovery. It is easy to extend these nonconvex penalty functions on singular values of a matrix to enhance low-rank matrix recovery. However, different from convex optimization, solving the nonconvex low-rank minimization problem is much more challenging than the nonconvex sparse minimization problem. We observe that all the existing nonconvex penalty functions are concave and monotonically increasing on [0, ∞). Thus their gradients are decreasing functions. Based on this property, we propose an Iteratively Reweighted Nuclear Norm (IRNN) algorithm to solve the nonconvex nonsmooth low-rank minimization problem. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem. By setting the weight vector as the gradient of the concave penalty function, the WSVT problem has a closed form solution. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthetic data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms.

Human pose estimation is a key step to action recognition. We propose a method of estimating 3D human poses from a single image, which works in conjunction with an existing 2D pose/joint detector. 3D pose estimation is challenging because multiple 3D poses may correspond to the same 2D pose after projection due to the lack of depth information. Moreover, current 2D pose estimators are usually inaccurate which may cause errors in the 3D estimation. We address the challenges in three ways: (i) We represent a 3D pose as a linear combination of a sparse set of bases learned from 3D human skeletons. (ii) We enforce limb length constraints to eliminate anthropomorphically implausible skeletons. (iii) We estimate a 3D pose by minimizing the 1-norm error between the projection of the 3D pose and the corresponding 2D detection. The 1-norm loss term is robust to inaccurate 2D joint estimations. We use the alternating direction method (ADM) to solve the optimization problem efficiently. Our approach outperforms the state-of-the-arts on three benchmark datasets.

In all feature point based partial-duplicate image retrieval systems, false matching is a common issue. To tackle the problem, geometric contexts are widely applied to filter the inconsistent matches. This paper presents a novel method called ℓ1-norm global geometric consistency. We first form the squared distance matrices of all the matched feature points, which remain invariant under translation and rotation between partial-duplicated images. Then we find the scale difference by solving a one-variable ℓ1-norm error minimization problem, where the large sparse errors correspond to the locations of inconsistent matches. By adopting the Golden Section Search method the minimization problem can be solved efficiently. Extensive experimental results show that our method reaches higher precisions than state-of-the-art geometric verification methods in detecting inconsistent matches. Its speed is also highly competitive even when compared to local geometric consistency based methods.

All existing feature point based partial-duplicate image retrieval systems are confronted with the false feature point matching problem. To resolve this issue, geometric contexts are widely used to verify the geometric consistency in order to remove false matches. However, most of the existing methods focus on local geometric contexts rather than global. Seeking global contexts has attracted a lot of attention in recent years. This paper introduces a novel global geometric consistency, based on the low rankness of squared distance matrices of feature points, to detect false matches. We cast the problem of detecting false matches as a problem of decomposing a squared distance matrix into a low rank matrix, which models the global geometric consistency, and a sparse matrix, which models the mismatched feature points. So we arrive at a model of Robust Principal Component Analysis. Our Low Rank Global Geometric Consistency (LRGGC) is simple yet effective and theoretically sound. Extensive experimental results show that our LRGGC is much more accurate than state of the art geometric verification methods in detecting false matches and is robust to all kinds of similarity transformation (scaling, rotation, and translation) and even slight change in 3D views. Its speed is also highly competitive even compared with local geometric consistency based methods.

Subspace clustering groups a set of samples (vectors) into clusters by approximating this set with a mixture of several linear subspaces, so that the samples in the same cluster are drawn from the same linear subspace. In majority of existing works on subspace clustering, samples are simply regarded as being independent and identically distributed, that is, arbitrarily ordering samples when necessary. However, this setting ignores sample correlations in their original spatial structure. To address this issue, we propose a tensor low-rank representation (TLRR) for subspace clustering by keeping available spatial information of data. TLRR seeks a lowest-rank representation over all the candidates while maintaining the inherent spatial structures among samples, and the affinity matrix used for spectral clustering is built from the combination of similarities along all data spatial directions. TLRR better captures the global structures of data and provides a robust subspace segmentation from corrupted data. Experimental results on both synthetic and real-world datasets show that TLRR outperforms several established state-of-the-art methods.

Geodesic distance, as an essential measurement for data dissimilarity, has been successfully used in manifold learning. However, most geodesic distance-based manifold learning algorithms have two limitations when applied to classification: 1) class information is rarely used in computing the geodesic distances between data points on manifolds and 2) little attention has been paid to building an explicit dimension reduction mapping for extracting the discriminative information hidden in the geodesic distances. In this paper, we regard geodesic distance as a kind of kernel, which maps data from linearly inseparable space to linear separable distance space. In doing this, a new semisupervised manifold learning algorithm, namely regularized geodesic feature learning algorithm, is proposed. The method consists of three techniques: a semisupervised graph construction method, replacement of original data points with feature vectors which are built by geodesic distances, and a new semisupervised dimension reduction method for feature vectors. Experiments on the MNIST, USPS handwritten digit data sets, MIT CBCL face versus nonface data set, and an intelligent traffic data set show the effectiveness of the proposed algorithm.

A novel discriminant analysis criterion is derived in this paper under the theoretical framework of Bayes optimality. In contrast to the conventional Fisher's discriminant criterion, the major novelty of the proposed one is the use of L1 norm rather than L2 norm, which makes it less sensitive to the outliers. With the L1-norm discriminant criterion, we propose a new linear discriminant analysis (L1-LDA) method for linear feature extraction problem. To solve the L1-LDA optimization problem, we propose an efficient iterative algorithm, in which a novel surrogate convex function is introduced such that the optimization problem in each iteration is to simply solve a convex programming problem and a close-form solution is guaranteed to this problem. Moreover, we also generalize the L1-LDA method to deal with the nonlinear robust feature extraction problems via the use of kernel trick, and hereafter proposed the L1-norm kernel discriminant analysis (L1-KDA) method. Extensive experiments on simulated and real data sets are conducted to evaluate the effectiveness of the proposed method in comparing with the state-of-the-art methods.

Many problems in statistics and machine learning (e.g., probabilistic graphical model, feature extraction, clustering and classification, etc) can be (re)formulated as linearly constrained separable convex programs. The traditional alternating direction method (ADM) or its linearized version (LADM) is for the two-variable case and \emphcannot be naively generalized to solve the multi-variable case. In this paper, we propose LADM with parallel splitting and adaptive penalty (LADMPSAP) to solve multi-variable separable convex programs efficiently. When all the component objective functions have bounded subgradients, we obtain convergence results that are stronger than those of ADM and LADM, e.g., allowing the penalty parameter to be unbounded and proving the \emphsufficient and necessary conditions for global convergence. We further propose a simple optimality measure and reveal the convergence \emphrate of LADMPSAP in an ergodic sense. For programs with extra convex set constraints, we devise a practical version of LADMPSAP for faster convergence. LADMPSAP is particularly suitable for sparse representation and low-rank recovery problems because its subproblems have closed form solutions and the sparsity and low-rankness of the iterates can be preserved during the iteration. It is also \emphhighly parallelizable and hence fits for parallel or distributed computing. Numerical experiments testify to the speed and accuracy advantages of LADMPSAP.

This paper studies the subspace segmentation problem. Given a set of data points drawn from a union of subspaces, the goal is to partition them into their underlying subspaces they were drawn from. The spectral clustering method is used as the framework. It requires to find an affinity matrix which is close to block diagonal, with nonzero entries corresponding to the data point pairs from the same subspace. In this work, we argue that both sparsity and the grouping effect are important for subspace segmentation. A sparse affinity matrix tends to be block diagonal, with less connections between data points from different subspaces. The grouping effect ensures that the highly corrected data which are usually from the same subspace can be grouped together. Sparse Subspace Clustering (SSC), by using ell^1-minimization, encourages sparsity for data selection, but it lacks of the grouping effect. On the contrary, Low-Rank Representation (LRR), by rank minimization, and Least Squares Regression (LSR), by ell^2-regularization, exhibit strong grouping effect, but they are short in subset selection. Thus the obtained affinity matrix is usually very sparse by SSC, yet very dense by LRR and LSR. In this work, we propose the Correlation Adaptive Subspace Segmentation (CASS) method by using trace Lasso. CASS is a data correlation dependent method which simultaneously performs automatic data selection and groups correlated data together. It can be regarded as a method which adaptively balances SSC and LSR. Both theoretical and experimental results show the effectiveness of CASS.

In this paper, we study the robust subspace clustering problem, which aims to cluster the given possibly noisy data points into their underlying subspaces. A large pool of previous subspace clustering methods focus on the graph construction by different regularization of the representation coefficient. We instead focus on the robustness of the model to non-Gaussian noises. We propose a new robust clustering method by using the correntropy induced metric, which is robust for handling the non-Gaussian and impulsive noises. Also we further extend the method for handling the data with outlier rows/features. The multiplicative form of half-quadratic optimization is used to optimize the non-convex correntropy objective function of the proposed models. Extensive experiments on face datasets well demonstrate that the proposed methods are more robust to corruptions and occlusions.

Character rectification is very important for character recognition. Front view standard character images are much easier to recognize since most character recognition algorithms were trained with such data. However, the existing text rectification methods only work for a paragraph or a page. We discover that the modified TILT algorithm can be applied to rectify many single Chinese, English, and digit characters robustly. By changing the character image into a low-rank texture image via binarization and gray level inversion, the modified TILT method applies a rank minimization technique to recover the deformation and the proposed algorithm can work for almost all characters. To further enhance the robustness of the proposed algorithm, the modified TILT algorithm is extended for short phrases that consist of multiple characters. Extensive experiments testify to the effectiveness of the proposed method in rectifying texts with significant affine or perspective deformation in real images, such as street signs taken by mobile phones.

In this paper, we address the subspace clustering problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to cluster the samples into their respective subspaces and remove possible outliers as well. To this end, we propose a novel objective function named Low-Rank Representation (LRR), which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that the convex program associated with LRR solves the subspace clustering problem in the following sense: When the data is clean, we prove that LRR exactly recovers the true subspace structures; when the data are contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for data corrupted by arbitrary sparse errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace clustering and error correction in an efficient and effective way.

Subspace clustering and feature extraction are two of the most commonly used unsupervised learning techniques in computer vision and pattern recognition. State-of-the-art techniques for subspace clustering make use of recent advances in sparsity and rank minimization. However, existing techniques are computationally expensive and may result in degenerate solutions that degrade clustering performance in the case of insufficient data sampling. To partially solve these problems, and inspired by existing work on matrix factorization, this paper proposes fixed-rank representation (FRR) as a unified framework for unsupervised visual learning. FRR is able to reveal the structure of multiple subspaces in closed-form when the data is noiseless. Furthermore, we prove that under some suitable conditions, even with insufficient observations, FRR can still reveal the true subspace memberships. To achieve robustness to outliers and noise, a sparse regularizer is introduced into the FRR framework. Beyond subspace clustering, FRR can be used for unsupervised feature extraction. As a non-trivial byproduct, a fast numerical solver is developed for FRR. Experimental results on both synthetic data and real applications validate our theoretical analysis and demonstrate the benefits of FRR for unsupervised visual learning.

Constructing a good graph to represent data structures is critical for many important machine learning tasks such as clustering and classification. This paper proposes a novel non-negative low-rank and sparse (NNLRS) graph for semi-supervised learning. The weights of edges in the graph are obtained by seeking a nonnegative low-rank and sparse matrix that represents each data sample as a linear combination of others. The so-obtained NNLRS-graph can capture both the global mixture of subspaces structure (by the low rankness) and the locally linear structure (by the sparseness) of the data, hence is both generative and discriminative. We demonstrate the effectiveness of NNLRS-graph in semi-supervised classification and discriminative analysis. Extensive experiments testify to the significant advantages of NNLRS-graph over graphs obtained through conventional means.

Manifold learning is an important feature extraction approach in data mining. This paper presents a new semi-supervised manifold learning algorithm, called Multi-Manifold Discriminative Analysis (Multi-MDA). The proposed method is designed to explore the discriminative information hidden in geodesic distances. The main contributions of the proposed method are: 1) we propose a semi-supervised graph construction method which can effectively capture the multiple manifolds structure of the data, 2) each data point is replaced with an associated feature vector whose elements are the graph distances from it to the other data points. Information of the nonlinear structure is contained in the feature vectors which are helpful for classification, 3) we propose a new semi-supervised linear dimension reduction method for feature vectors which introduces the class information into the manifold learning process and establishes an explicit dimension reduction mapping. Experiments on benchmark data sets are conducted to show the effectiveness of the proposed method.

Many machine learning and signal processing problems can be formulated as linearly constrained convex programs, which could be efficiently solved by the alternating direction method (ADM). However, usually the subproblems in ADM are easily solvable only when the linear mappings in the constraints are identities. To address this issue, we propose a linearized ADM (LADM) method by linearizing the quadratic penalty term and adding a proximal term when solving the subproblems. For fast convergence, we also allow the penalty to change adaptively according a novel update rule. We prove the global convergence of LADM with adaptive penalty (LADMAP). As an example, we apply LADMAP to solve low-rank representation (LRR), which is an important subspace clustering technique yet suffers from high computation cost. By combining LADMAP with a skinny SVD representation technique, we are able to reduce the complexity $O(n^3)$ of the original ADM based method to $O(rn^2)$, where $r$ and $n$ are the rank and size of the representation matrix, respectively, hence making LRR possible for large scale applications. Numerical experiments verify that for LRR our LADMAP based methods are much faster than state-of-the-art algorithms.

We present a novel approach to reconstruction-based super-resolution that uses aperiodic pixel tilings, such as a Penrose tiling or a biological retina, for improved performance. To this aim, we develop a new variant of the well-known error back projection super-resolution algorithm that makes use of the exact detector model in its back projection operator for better accuracy. Pixels in our model can vary in shape and size, and there may be gaps between adjacent pixels. The algorithm applies equally well to periodic or aperiodic pixel tilings. We present analysis and extensive tests using synthetic and real images to show that our approach using aperiodic layouts substantially outperforms existing reconstruction-based algorithms for regular pixel arrays. We close with a discussion of the feasibility of manufacturing CMOS or CCD chips with pixels arranged in Penrose tilings.

A color filter array (CFA) used in a digital camera is a mosaic of spectrally selective filters, which allows only one color component to be sensed at each pixel. The missing two components of each pixel have to be estimated by methods known as demosaicking. The demosaicking algorithm and the CFA design are crucial for the quality of the output images. In this paper, we present a CFA design methodology in the frequency domain. The frequency structure, which is shown to be just the symbolic DFT of the CFA pattern (one period of the CFA), is introduced to represent images sampled with any rectangular CFAs in the frequency domain. Based on the frequency structure, the CFA design involves the solution of a constrained optimization problem that aims at minimizing the demosaicking error. To decrease the number of parameters and speed up the parameter searching, the optimization problem is reformulated as the selection of geometric points on the boundary of a convex polygon or the surface of a convex polyhedron. Using our methodology, several new CFA patterns are found, which outperform the currently commercialized and published ones. Experiments demonstrate the effectiveness of our CFA design methodology and the superiority of our new CFA patterns.

This paper proposes a generalized accelerated proximal gradient (GAPG) approach for solving total variation (TV)-based image restoration problems. The GAPG algorithm generalizes the original APG algorithm by replacing the Lipschitz constant with an appropriate positive-definite matrix, resulting in faster convergence. For TV-based image restoration problems, we further introduce two auxiliary variables that approximate the partial derivatives. Constraints on the variables can easily be imposed without modifying the algorithm much, and the TV regularization can be either isotropic or anisotropic. As compared with the recently developed APG-based methods for TV-based image restoration, i.e., monotone version of the two-step iterative shrinkage/thresholding algorithm (MTwIST) and monotone version of the fast IST algorithm (MFISTA), our GAPG is much simpler as it does not require to solve an image denoising subproblem. Moreover, the convergence rate of O(k(-2)) is maintained by our GAPG, where k is the number of iterations; the cost of each iteration in GAPG is also lower. As a result, in our experiments, our GAPG approach can be much faster than MTwIST and MFISTA. The experiments also verify that our GAPG converges faster than the original APG and MTwIST when they solve identical problems.

Discriminant feature extraction plays a central role in pattern recognition and classification. Linear Discriminant Analysis (LDA) is a traditional algorithm for supervised feature extraction. Recently, unlabeled data have been utilized to improve LDA. However, the intrinsic problems of LDA still exist and only the similarity among the unlabeled data is utilized. In this paper, we propose a novel algorithm, called Semisupervised Semi-Riemannian Metric Map (S3RMM), following the geometric framework of semi Riemannian manifolds. S3RMM maximizes the discrepancy of the separability and similarity measures of scatters formulated by using semi-Riemannian metric tensors. The metric tensor of each sample is learned via semisupervised regression. Our method can also be a general framework for proposing new semisupervised algorithms, utilizing the existing discrepancy-criterion-based algorithms. The experiments demonstrated on faces and handwritten digits show that S3RMM is promising for semisupervised feature extraction.

Palm lines are the most important features for palmprint recognition. They are best considered as typical multiscale features, where the principal lines can be represented at a larger scale while the wrinkles at a smaller scale. Motivated by the success of coding-based palmprint recognition methods, this paper investigates a compact representation of multiscale palm line orientation features, and proposes a novel method called the sparse multiscale competitive code (SMCC). The SMCC method first defines a filter bank of second derivatives of Gaussians with different orientations and scales, and then uses the l1-norm sparse coding to obtain a robust estimation of the multiscale orientation field. Finally, a generalized competitive code is used to encode the dominant orientation. Experimental results show that the SMCC achieves higher verification accuracy than state-of-the-art palmprint recognition methods, yet uses a smaller template size than other multiscale methods.

This paper proposes a new approach to theoretically analyze compressive sensing directly from the randomly sampling matrix phi instead of a certain recovery algorithm. For simplifying our analyses, we assume both input source and random sampling matrix as binary. Taking anyone of source bits, we can constitute a tree by parsing the randomly sampling matrix, where the selected source bit as the root. In the rest of tree, measurement nodes and source nodes are connected alternatively according to phi. With the tree, we can formulate the probability if one source bit can be recovered from randomly sampling measurements. The further analyses upon the tree structure reveal the relation between the un-recovery probability with random measurements and the un-recovery probability with source sparsity. The conditions of successful recovery are proven on the parameter S-M plane. Then the results of the tree structure based analyses are compared with the actual recovery process.

In this work, we address the problem of performing class-specific unsupervised object segmentation, i.e., automatic segmentation without annotated training images. Object segmentation can be regarded as a special data clustering problem where both class-specific information and local texture/color similarities have to be considered. To this end, we propose a hybrid graph model (HGM) that can make effective use of both symmetric and asymmetric relationship among samples. The vertices of a hybrid graph represent the samples and are connected by directed edges and/or undirected ones, which represent the asymmetric and/or symmetric relationship between them, respectively. When applied to object segmentation, vertices are superpixels, the asymmetric relationship is the conditional dependence of occurrence, and the symmetric relationship is the color/texture similarity. By combining the Markov chain formed by the directed subgraph and the minimal cut of the undirected subgraph, the object boundaries can be determined for each image. Using the HGM, we can conveniently achieve simultaneous segmentation and recognition by integrating both top-down and bottom-up information into a unified process. Experiments on 42 object classes (9,415 images in total) show promising results.

Discriminant analysis plays an important role in statistical pattern recognition. A popular method is the Foley-Sammon optimal discriminant vectors (FSODVs) method, which aims to find an optimal set of discriminant vectors that maximize the Fisher discriminant criterion under the orthogonal constraint. The FSODVs method outperforms the classic Fisher linear discriminant analysis (FLDA) method in the sense that it can solve more discriminant vectors for recognition. Kernel Foley-Sammon optimal discriminant vectors (KFSODVs) is a nonlinear extension of FSODVs via the kernel trick. However, the current KFSODVs algorithm may suffer from the heavy computation problem since it involves computing the inverse of matrices when solving each discriminant vector, resulting in a cubic complexity for each discriminant vector. This is costly when the number of discriminant vectors to be computed is large. In this paper, we propose a fast algorithm for solving the KFSODVs, which is based on rank-one update (ROU) of the eigensytems. It only requires a square complexity for each discriminant vector. Moreover, we also generalize our method to efficiently solve a family of optimally constrained generalized Rayleigh quotient (OCGRQ) problems which include many existing dimensionality reduction techniques. We conduct extensive experiments on several real data sets to demonstrate the effectiveness of the proposed algorithms.

This paper proposes to use a bipartite graph to represent compressive sensing (CS). The evolution of nodes and edges in the bipartite graph, which is equivalent to the decoding process of compressive sensing, is characterized by a set of differential equations. One of main contributions in this paper is that we derive the close-form formulation of the evolution in statistics, which enable us to more accurately analyze the performance of compressive sensing. Based on the formulation, the distortion of random sampling and the rate needed to code measurements are analyzed briefly. Finally, numerical experiments verify our formulation of the evolution and the rate-distortion curves of compressive sensing are drawn to be compared with entropy coding.

Non-frontal view facial expression recognition is important in many scenarios where the frontal view face images may not be available. However, few work on this issue has been done in the past several years because of its technical challenges and the lack of appropriate databases. Recently, a 3D facial expression database (BU-3DFE database) is collected by Yin et al. [10] and has attracted some researchers to study this issue. Based on the BU-3DFE database, in this paper we propose a novel approach to expression recognition from non-frontal view facial images. The novelty of the proposed method lies in recognizing the multi-view expressions under the unified Bayes theoretical framework, where the recognition problem can be formulated as an optimization problem of minimizing an upper bound of Bayes error. We also propose a close-form solution method based on the power iteration approach and rank-one update (ROU) technique to find the optimal solutions of the proposed method. Extensive experiments on BU-3DFE database with 100 subjects and 5 yaw rotation view angles demonstrate the effectiveness of our method.

The method of common spatio-spectral patterns (CSSPs) is an extension of common spatial patterns (CSPs) by utilizing the technique of delay embedding to alleviate the adverse effects of noises and artifacts on the electroencephalogram (EEG) classification. Although the CSSPs method has shown to be more powerful than the CSPs method in the EEG classification, this method is only suitable for two-class EEG classification problems. In this paper, we generalize the two-class CSSPs method to multi-class cases. To this end, we first develop a novel theory of multi-class Bayes error estimation and then present the multi-class CSSPs (MCSSPs) method based on this Bayes error theoretical framework. By minimizing the estimated closed-form Bayes error, we obtain the optimal spatio-spectral filters of MCSSPs. To demonstrate the effectiveness of the proposed method, we conduct extensive experiments on the data set of BCI competition 2005. The experimental results show that our method significantly outperforms the previous multi-class CSPs (MCSPs) methods in the EEG classification.

We present a simple new criterion for classification, based on principles from lossy data compression. The criterion assigns a test sample to the class that uses the minimum number of additional bits to code the test sample, subject to an allowable distortion. We demonstrate the asymptotic optimality of this criterion for Gaussian distributions and analyze its relationships to classical classifiers. The theoretical results clarify the connections between our approach and popular classifiers such as maximum a posteriori (MAP), regularized discriminant analysis (RDA), k-nearest neighbor (k-NN), and support vector machine (SVM), as well as unsupervised methods based on lossy coding. Our formulation induces several good effects on the resulting classifier. First, minimizing the lossy coding length induces a regularization effect which stabilizes the (implicit) density estimate in a small sample setting. Second, compression provides a uniform means of handling classes of varying dimension. The new criterion and its kernel and local versions perform competitively on synthetic examples, as well as on real imagery data such as handwritten digits and face images. On these problems, the performance of our simple classifier approaches the best reported results, without using domain-specific information. All MATLAB code and classification results are publicly available for peer evaluation at http://perception.csl.uiuc.edu/coding/home.htm.

In this paper, we aim to handle the intraclass variation resulting from the geometric transformation and the illumination change for more robust texture classification. To this end, we propose a novel feature descriptor called Radon representation-based feature descriptor (RRFD). RRFD converts the original pixel represented images into Radon-pixel images by using the Radon transform. The new Radon-pixel representation is more informative in geometry and has a much lower dimension. Subsequently, RRFD efficiently achieves affine invariance by projecting an image (or an image patch) from the space of Radon-pixel pairs onto an invariant feature space by using a ratiogram, i.e., the histogram of ratios between the areas of triangle pairs. The illumination invariance is also achieved by defining an illumination invariant distance metric on the invariant feature space. Comparing to the existing Radon transform-based texture features, which only achieve rotation and/or scaling invariance, RRFD achieves affine invariance. The experimental results on CUReT show that RRFD is a powerful feature descriptor that is suitable for texture classification.

We propose a kernel Nyström method for reconstructing the light transport matrix from a relatively small number of acquired images. Our work is based on the generalized Nyström method for low rank matrices. We introduce the light transport kernel and incorporate it into the Nyström method to exploit the nonlinear coherence of the light transport matrix. We also develop an adaptive scheme for efficiently capturing the sparsely sampled images from the scene. Our experiments indicate that the kernel Nyström method can achieve good reconstruction of the light transport matrix with a few hundred images and produce high quality relighting results. The kernel Nyström method is effective for modeling scenes with complex lighting effects and occlusions which have been challenging for existing techniques.

In this paper, we develop a geometric framework for linear or nonlinear discriminant subspace learning and classification. In our framework, the structures of classes are conceptualized as a semi-Riemannian manifold which is considered as a submanifold embedded in an ambient semi-Riemannian space. The class structures of original samples can be characterized and deformed by local metrics of the semi-Riemannian space. Semi-Riemannian metrics are uniquely determined by the smoothing of discrete functions and the nullity of the semi-Riemannian space. Based on the geometrization of class structures, optimizing class structures in the feature space is equivalent to maximizing the quadratic quantities of metric tensors in the semi-Riemannian space. Thus supervised discriminant subspace learning reduces to unsupervised semi-Riemannian manifold learning. Based on the proposed framework, a novel algorithm, dubbed as Semi-Riemannian Discriminant Analysis (SRDA), is presented for subspace-based classification. The performance of SRDA is tested on face recognition (singular case) and handwritten capital letter classification (nonsingular case) against existing algorithms. The experimental results show that SRDA works well on recognition and classification, implying that semi-Riemannian geometry is a promising new tool for pattern recognition and machine learning.

In this article, we propose techniques for modeling and rendering of heterogeneous translucent materials that enable acquisition from measured samples, interactive editing of material attributes, and real-time rendering. The materials are assumed to be optically dense such that multiple scattering can be approximated by a diffusion process described by the diffusion equation. For modeling heterogeneous materials, we present the inverse diffusion algorithm for acquiring material properties from appearance measurements. This modeling algorithm incorporates a regularizer to handle the ill-conditioning of the inverse problem, an adjoint method to dramatically reduce the computational cost, and a hierarchical GPU implementation for further speedup. To render an object with known material properties, we present the polygrid diffusion algorithm , which solves the diffusion equation with a boundary condition defined by the given illumination environment. This rendering technique is based on representation of an object by a polygrid, a grid with regular connectivity and an irregular shape, which facilitates solution of the diffusion equation in arbitrary volumes. Because of the regular connectivity, our rendering algorithm can be implemented on the GPU for real-time performance. We demonstrate our techniques by capturing materials from physical samples and performing real-time rendering and editing with these materials.

Discriminant feature extraction plays a fundamental role in pattern recognition. In this paper, we propose the linear Laplacian discrimination (LLD) algorithm/or discriminant feature extraction. LLD is an extension of linear discriminant analysis (LDA). Our motivation is to address the issue that LDA cannot work well in cases where sample spaces are non-Euclidean. Specifically, we define the within-class scatter and the between-class scatter using similarities which are based on pairwise distances in sample spaces. Thus the structural information of classes is contained in the within-class and the between-class Laplacian matrices which are free from metrics of sample spaces. The optimal discriminant subspace can be derived by controlling the structural evolution of Laplacian matrices. Experiments are performed on the facial database for FRGC version 2. Experimental results show that LLD is effective in extracting discriminant features.

We present a novel approach to reconstruction based super- resolution that explicitly models the detector's pixel layout. Pixels in our model can vary in shape and size, and there may be gaps between adjacent pixels. Furthermore, their layout can be periodic as well as aperiodic, such as penrose tiling or a biological retina. We also present a new variant of the well known error back-projection super-resolution algorithm that makes use of the exact detector model in its back projection operator for better accuracy. Our method can be applied equally well to either periodic or aperiodic pixel tiling. Through analysis and extensive testing using synthetic and real images, we show that our approach outperforms existing reconstruction based algorithms for regular pixel arrays. We obtain significantly better results using aperiodic pixel layouts. As an interesting example, we apply our method to a retina-like pixel structure modeled by a centroidal Voronoi tessellation. We demonstrate that, in principle, this structure is better for super-resolution than the regular pixel array used in today's sensors.

Learning-based superresolution (SR) are popular SR techniques that use application dependent priors to infer the missing details in low resolution images (LRIs). However, their performance still deteriorates quickly when the magnification factor is moderately large. This leads us to an important problem: "Do limits of learning-based SR algorithms exist?" In this paper, we attempt to shed some light on this problem when the SR algorithms are designed for general natural images (GNIs). We first define an expected risk for the SR algorithms that is based on the root mean squared error between the superresolved images and the ground truth images. Then utilizing the statistics of GNIs, we derive a closed form estimate of the lower bound of the expected risk. The lower bound can be computed by sampling real images. By computing the curve of the lower bound w.r.t. the magnification factor, we can estimate the limits of learning-based SR algorithms, at which the lower bound of expected risk exceeds a relatively large threshold. We also investigate the sufficient number of samples to guarantee an accurate estimation of the lower bound.

Structural perception of data plays a fundamental role in pattern analysis and machine learning. In this paper, we develop a new structural perception of data based on local contexts. We first identify the contextual set of a point by finding its nearest neighbors. Then the contextual distance between the point and one of its neighbors is defined by the difference between their contribution to the integrity of the geometric structure of the contextual set, which is depicted by a structural descriptor. The centroid and the coding length are introduced as the examples of descriptors of the contextual set. Furthermore, a directed graph (digraph) is built to model the asymmetry of perception. The edges of the digraph are weighted based on the contextual distances. Thus direction is brought to the undirected data. And the structural perception of data can be performed by mining the properties of the digraph. We also present the method for deriving the global digraph Laplacian from the alignment of the local digraph Laplacians. Experimental results on clustering and ranking of toy problems and real data show the superiority of asymmetric perception.

Dimensionality reduction plays a fundamental role in data processing, for which principal component analysis (PCA) is widely used. In this paper, we develop the Laplacian PCA (LPCA) algorithm which is the extension of PCA to a more general form by locally optimizing the weighted scatter. In addition to the simplicity of PCA, the benefits brought by LPCA are twofold: the strong robustness against noise and the weak metric-dependence on sample spaces. The LPCA algorithm is based on the global alignment of locally Gaussian or linear subspaces via an alignment technique borrowed from manifold learning. Based on the coding length of local samples, the weights can be determined to capture the local principal structure of data. We also give the exemplary application of LPCA to manifold learning. Manifold unfolding (non-linear dimensionality reduction) can be performed by the alignment of tangential maps which are linear transformations of tangent coordinates approximated by LPCA. The superiority of LPCA to PCA and kernel PCA is verified by the experiments on face recognition (FRGC version 2 face database) and manifold (Scherk surface) unfolding.

We present a simple new criterion for classification, based on principles from lossy data compression. The criterion assigns a test sample to the class that uses the min- imum number of additional bits to code the test sample, subject to an allowable distortion. We prove asymptotic optimality of this criterion for Gaussian data and analyze its relationships to classical classifiers. Theoretical results provide new insights into relationships among popular classifiers such as MAP and RDA, as well as unsupervised clustering methods based on lossy compression [13]. Mini- mizing the lossy coding length induces a regularization effect which stabilizes the (implicit) density estimate in a small-sample setting. Compression also provides a uniform means of handling classes of varying dimension. This simple classi- fication criterion and its kernel and local versions perform competitively against existing classifiers on both synthetic examples and real imagery data such as hand- written digits and human faces, without requiring domain-specific information.

Current techniques for generating animated scenes involve either videos (whose resolution is limited) or a single image (which requires a significant amount of user interaction). In this paper, we describe a system that allows the user to quickly and easily produce a compelling-looking animation from a small collection of high resolution stills. Our system has two unique features. First, it applies an automatic partial temporal order recovery algorithm to the stills in order to approximate the original scene dynamics. The output sequence is subsequently extracted using a second-order Markov Chain model. Second, a region with large motion variation can be automatically decomposed into semiautonomous regions such that their temporal orderings are softly constrained. This is to ensure motion smoothness throughout the original region. The final animation is obtained by frame interpolation and feathering. Our system also provides a simple-to-use interface to help the user to fine-tune the motion of the animated scene. Using our system, an animated scene can be generated in minutes. We show results for a variety of scenes.

In documents, tables are important structured objects that present statistical and relational information. In this paper, we present a robust system which is capable of detecting tables from free style online ink notes and extracting their structure so that they can be further edited in multiple ways. First, the primative structure of tables, i.e., candidates for ruling lines and table bounding boxes, are detected among drawing strokes. Second, the logical structure of tables is determined by normalizing the table skeletons, identifying the skeleton structure, and extracting the cell contents. The detection process is similar to a decision tree so that invalid candidates can be ruled out quickly. Experimental results suggest that our system is robust and accurate in dealing with tables having complex structure or drawn under complex situations.

Wang and Feng (IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 5, p 846, May 2006) pointed out that the deduction in (Z. Lin and H. Y. Shum, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 1, pp. 83-97, Jan. 2004) overlooked the validity of the perturbation theorem used in (Z. Lin and H. Y. Shum, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 1, pp. 83-97, Jan. 2004). In this paper, we show that, when the perturbation theorem is invalid, the probability of successful superresolution is very low. Therefore, we only have to derive the limits under the condition that validates the perturbation theorem, as done in (Z. Lin and H. Y. Shum, IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 1, pp. 83-97, Jan. 2004).

The advance in image/video editing techniques has facilitated people in synthesizing realistic images/videos that may hard to be distinguished from real ones by visual examination. This poses a problem: how to differentiate real images/videos from doctored ones? This is a serious problem because some legal issues may occur if there is no reliable way for doctored image/video detection when human inspection fails. Digital watermarking cannot solve this problem completely. We propose an approach that computes the response functions of the camera by selecting appropriate patches in different ways. An image may be doctored if the response functions are abnormal or inconsistent to each other. The normality of the response functions is classified by a trained support vector machine (SVM). Experiments show that our method is effective for high-contrast images with many textureless edges.

Super-resolution is a technique that produces higher resolution images from low resolution images (LRIs). In practice, the improvement in resolution is limited. The aim of this paper is to address the problem of whether fundamental limits exist for super-resolution? Specifically, this paper provides explicit limits for a major class of super-resolution algorithms, called reconstruction-based algorithms, under both real and synthetic conditions. Our analysis is based on perturbation theory of linear systems. We also show that a sufficient number of LRIs can be determined to reach the limit. Both real and synthetic experiments are carried out to verify our analysis.

Image-based relighting (IBL) is a technique to change the illumination of an image-based object/scene. In this paper, we define a representation called the reflected irradiance field which records the reflection from an object surface irradiated by a point light source that moves on a plane. This representation is dual to that of the light field. It synthesizes a novel image under a different illumination by interpolating and superimposing appropriate recorded samples. Furthermore, we study the minimum sampling problem of the reflected irradiance field, i.e., how many point light sources are needed during sampling. We find that there exists a geometry-independent bound for the sampling interval whenever the second-order derivatives of the surface BRDF and the minimum depth of the scene are bounded. This bound ensures that the error in the reconstructed image is controlled by a given tolerance, regardless of the geometry. Experiments on both synthetic and real surfaces are conducted to verify our analysis.

While several image-based rendering techniques have been proposed to successfully render scenes/objects from a large collection (e.g., thousands) of images without explicitly recovering 3D structures, the minimum number of images needed to achieve a satisfactory rendering result remains an open problem. This paper is the first attempt to investigate the lower bound for the number of samples needed in the Lumigraph/light field rendering. To simplify the analysis, we consider an ideal scene with only a point that is between a minimum and a maximum range. Furthermore, constant-depth assumption and bilinear interpolation are used for rendering. The constant-depth assumption serves to choose "nearby" rays for interpolation. Our criterion to determine the lower bound is to avoid horizontal and vertical double images, which are caused by interpolation using multiple nearby rays. This criterion is based on the causality requirement in scale-space theory, i.e., no "spurious details" should be generated while smoothing. Using this criterion, closed-form solutions of lower bounds are obtained for both 3D plenoptic function (Concentric Mosaics) and 4D plenoptic function (light field). The bounds are derived completely from the aspect of geometry and are closely related to the resolution of the camera and the depth range of the scene. These tower bounds are further verified by our experimental results.