Large language models in the past have typically relied on some form of reinforcement learning with human feedback (RLHF) to better align model responses with human preferences. However, because of oft-observed instabilities when implementing these RLHF pipelines, various reparameterization techniques have recently been introduced to sidestep the need for separately learning an RL reward model. Instead, directly fine-tuning for human preferences is achieved via the minimization of a single closed-form training objective, a process originally referred to as direct preference optimization (DPO). Although effective in certain real-world settings, we detail how the foundational role of DPO reparameterizations (and equivalency to applying RLHF with an optimal reward) may be obfuscated once inevitable optimization constraints are introduced during model training. This then motivates alternative derivations and analysis of DPO that remain intact even in the presence of such constraints. As initial steps in this direction, we re-derive DPO from a simple Gaussian estimation perspective, with strong ties to compressive sensing and classical constrained optimization problems involving noise-adaptive, concave regularization.

Prior-Fitted Networks (PFNs) have recently been proposed to efficiently perform tabular classification tasks. Although they achieve good performance on small datasets, they encounter limitations with larger datasets. These limitations include significant memory consumption and increased computational complexity, primarily due to the impracticality of incorporating all training samples as inputs within these networks. To address these challenges, we investigate the fitting assumption for PFNs and input samples. Building on this understanding, we propose \emph{BoostPFN} designed to enhance the performance of these networks, especially for large-scale datasets. We also theoretically validate the convergence of BoostPFN and our empirical results demonstrate that the BoostPFN method can outperform standard PFNs with the same size of training samples in large datasets and achieve a significant acceleration in training times compared to other established baselines in the field, including widely-used Gradient Boosting Decision Trees (GBDTs), deep learning methods and AutoML systems. High performance is maintained for up to 50x of the pre-training size of PFNs, substantially extending the limit of training samples. Through this work, we address the challenges of efficiently handling large datasets via PFN-based models, paving the way for faster and more effective tabular data classification training and prediction process.

Among the many variants of graph neural network (GNN) architectures capable of modeling data with cross-instance relations, an important subclass involves layers designed such that the forward pass iteratively reduces a graph-regularized energy function of interest. In this way, node embeddings produced at the output layer dually serve as both predictive features for solving downstream tasks (e.g., node classification) and energy function minimizers that inherit transparent, exploitable inductive biases and interpretability. However, scaling GNN architectures constructed in this way remains challenging, in part because the convergence of the forward pass may involve models with considerable depth. To tackle this limitation, we propose a sampling-based energy function and scalable GNN layers that iteratively reduce it, guided by convergence guarantees in certain settings. We also instantiate a full GNN architecture based on these designs, and the model achieves competitive accuracy and scalability when applied to the largest publicly-available node classification benchmark exceeding 1TB in size. Our source code is available at https://github.com/haitian-jiang/MuseGNN.

Modern language models have demonstrated remarkable reasoning capabilities by using chain-of-thought (CoT). One hypothesis about the inner workings of CoT is that it breaks down originally complex tasks into smaller subtasks that are more amenable to learning. We formalize this notion by showing possibility and impossibility results of learning from in-context demonstrations with and without CoT. In particular, with CoT, we examine a family of learning algorithms that learn a task step-by-step, capable of composing simpler functions from individual reasoning steps to form an overall complex function. This process reduces the difficulty of learning a task to that of the hardest reasoning step in the chain. Moreover, we prove Transformers can express this algorithm and thus they can efficiently in-context learn arbitrary tasks as long as these tasks can be decomposed into a finite number of subtasks, each of which are efficiently learnable. In contrast, without CoT, we demonstrate that there exist tasks that are inherently unlearnable by the same algorithm. Overall, our results suggest several provably effective ways for decomposing target problems to instantiate CoT. Empirically, we demonstrate our proposed CoT construction significantly enhances the reasoning capabilities of real-world LLMs in solving challenging arithmetic reasoning tasks, including learning polynomials and Boolean formulas.

The generated responses of large language models (LLMs) are often fine-tuned to human preferences through a process called reinforcement learning from human feedback (RLHF). As RLHF relies on a challenging training sequence, whereby a separate reward model is independently learned and then later applied to LLM policy updates, ongoing research effort has targeted more straightforward alternatives. In this regard, direct preference optimization (DPO) and its many offshoots circumvent the need for a separate reward training step. Instead, through the judicious use of a reparameterization trick that induces an implicit reward, DPO and related methods consolidate learning to the minimization of a single loss function. And yet despite demonstrable success in some real-world settings, we prove that DPO-based objectives are nonetheless subject to sub-optimal regularization and counter-intuitive interpolation behaviors, underappreciated artifacts of the reparameterizations upon which they are based. To this end, we introduce an explicit preference optimization framework termed EXPO that requires no analogous reparameterization to achieve an implicit reward. Quite differently, we merely posit intuitively-appealing regularization factors from scratch that transparently avoid the potential pitfalls of key DPO variants, provably satisfying regularization desiderata that prior methods do not. Empirical results serve to corroborate our analyses and showcase the efficacy of EXPO.

Is there really much more to say about sparse autoencoders (SAEs)? Autoencoders in general, and SAEs in particular, represent deep architectures that are capable of modeling low-dimensional latent structure in data. Such structure could reflect, among other things, correlation patterns in large language model activations, or complex natural image manifolds. And yet despite the wide-ranging applicability, there have been relatively few changes to SAEs beyond the original recipe from decades ago, namely, standard deep encoder/decoder layers trained with a classical/deterministic sparse regularizer applied within the latent space. One possible exception is the variational autoencoder (VAE), which adopts a stochastic encoder module capable of producing sparse representations when applied to manifold data. In this work we formalize underappreciated weaknesses with both canonical SAEs, as well as analogous VAEs applied to similar tasks, and propose a hybrid alternative model that circumvents these prior limitations. In terms of theoretical support, we prove that global minima of our proposed model recover certain forms of structured data spread across a union of manifolds. Meanwhile, empirical evaluations on synthetic and real-world datasets substantiate the efficacy of our approach in accurately estimating underlying manifold dimensions and producing sparser latent representations without compromising reconstruction error. In general, we are able to exceed the performance of equivalent-capacity SAEs and VAEs, as well as recent diffusion models where applicable, within domains such as images and language model activation patterns.

We introduce Griffin, the first foundation model attemptation designed specifically for Relational Databases (RDBs). Unlike previous smaller models focused on single RDB tasks, Griffin unifies the data encoder and task decoder to handle diverse tasks. Additionally, we enhance the architecture by incorporating a cross-attention module and a novel aggregator. Griffin utilizes pretraining on both single-table and RDB datasets, employing advanced encoders for categorical, numerical, and metadata features, along with innovative components such as cross-attention modules and enhanced message-passing neural networks (MPNNs) to capture the complexities of relational data. Evaluated on large-scale, heterogeneous, and temporal graphs extracted from RDBs across various domains (spanning over 150 million nodes), Griffin demonstrates superior or comparable performance to individually trained models, excels in low-data scenarios, and shows strong transferability with similarity and diversity in pretraining across new datasets and tasks, highlighting its potential as a universally applicable foundation model for RDBs. Code available at https://github.com/yanxwb/Griffin.

Learning representations for structured data with certain geometries (e.g., observed or unobserved) is a fundamental challenge, wherein message passing neural networks (MPNNs) have become a de facto class of model solutions. In this paper, inspired by physical systems, we propose an energy-constrained diffusion model, which integrates the inductive bias of diffusion on manifolds with layer-wise constraints of energy minimization. We identify that the diffusion operators have a one-to-one correspondence with the energy functions implicitly descended by the diffusion process, and the finite-difference iteration for solving the energy-constrained diffusion system induces the propagation layers of various types of MPNNs operating on observed or latent structures. This leads to a unified mathematical framework for common neural architectures whose computational flows can be cast as message passing (or its special case), including MLPs, GNNs, and Transformers. Building on these insights, we devise a new class of neural message passing models, dubbed diffusion-inspired Transformers (DIFFormer), whose global attention layers are derived from the principled energy-constrained diffusion framework. Across diverse datasets ranging from real-world networks to images, texts, and physical particles, we demonstrate that the new model achieves promising performance in scenarios where the data structures are observed (as a graph), partially observed, or entirely unobserved.

It has been observed that message-passing graph neural networks (GNN) sometimes struggle to maintain a healthy balance between the efficient / scalable modeling of long-range dependencies across nodes while avoiding unintended consequences such oversmoothed node representations, sensitivity to spurious edges, or inadequate model interpretability. To address these and other issues, two separate strategies have recently been proposed, namely implicit and unfolded GNNs (that we abbreviate to IGNN and UGNN respectively). The former treats node representations as the fixed points of a deep equilibrium model that can efficiently facilitate arbitrary implicit propagation across the graph with a fixed memory footprint. In contrast, the latter involves treating graph propagation as unfolded descent iterations as applied to some graph-regularized energy function. While motivated differently, in this paper we carefully quantify explicit situations where the solutions they produce are equivalent and others where their properties sharply diverge. This includes the analysis of convergence, representational capacity, and interpretability. In support of this analysis, we also provide empirical head-to-head comparisons across multiple synthetic and public real-world node classification benchmarks. These results indicate that while IGNN is substantially more memory-efficient, UGNN models support unique, integrated graph attention mechanisms and propagation rules that can achieve strong node classification accuracy across disparate regimes such as adversarially-perturbed graphs, graphs with heterophily, and graphs involving long-range dependencies.

Graph neural networks (GNNs) for link prediction can loosely be divided into two broad categories. First,node-wisearchitectures pre-compute individual embeddings for each node that are later combined by a simple decoder to make predictions. While extremely efficient at inference time, model expressiveness is limited such that isomorphic nodes contributing to candidate edges may not be distinguishable, compromising accuracy. In contrast,edge-wisemethods rely on the formation of edge-specific subgraph embeddings to enrich the representation of pair-wise relationships, disambiguating isomorphic nodes to improve accuracy, but with increased model complexity. To better navigate this trade-off, we propose a novel GNN architecture whereby theforward passexplicitly depends onbothpositive (as is typical) and negative (unique to our approach) edges to inform more flexible, yet still cheap node-wise embeddings. This is achieved by recasting the embeddings themselves as minimizers of a forward-pass-specific energy function that favors separation of positive and negative samples. Notably, this energy is distinct from the actual training loss shared by most existing link prediction models, where contrastive pairs only influence thebackward pass. As demonstrated by extensive empirical evaluations, the resulting architecture retains the inference speed of node-wise models, while producing competitive accuracy with edge-wise alternatives.

Bilevel optimization refers to scenarios whereby the optimal solution of a lower-level energy function serves as input features to an upper-level objective of interest. These optimal features typically depend on tunable parameters of the lower-level energy in such a way that the entire bilevel pipeline can be trained end-to-end. Although not generally presented as such, this paper demonstrates how a variety of graph learning techniques can be recast as special cases of bilevel optimization or simplifications thereof. In brief, building on prior work we first derive a more flexible class of energy functions that, when paired with various descent steps (e.g., gradient descent, proximal methods, momentum, etc.), form graph neural network (GNN) message-passing layers; critically, we also carefully unpack where any residual approximation error lies with respect to the underlying constituent message-passing functions. We then probe several simplifications of this framework to derive close connections with non-GNN-based graph learning approaches, including knowledge graph embeddings, various forms of label propagation, and efficient graph-regularized MLP models. And finally, we present supporting empirical results that demonstrate the versatility of the proposed bilevel lens, which we refer to as BloomGML, referencing that BiLevel Optimization Offers More Graph Machine Learning. Our code is available at \url{https://github.com/amberyzheng/BloomGML}. Let graph ML bloom.

Crafting effective features is a crucial yet labor-intensive and domain-specific task within machine learning pipelines. Fortunately, recent advancements in Large Language Models (LLMs) have shown promise in automating various data science tasks, including feature engineering. But despite this potential, evaluations thus far are primarily based on the end performance of a complete ML pipeline, providing limited insight into precisely how LLMs behave relative to human experts in feature engineering. To address this gap, we propose ELF-Gym, a framework for Evaluating LLM-generated Features. We curated a new dataset from historical Kaggle competitions, including 251 golden features used by top-performing teams. ELF-Gym then quantitatively evaluates LLM-generated features by measuring their impact on downstream model performance as well as their alignment with expert-crafted features through semantic and functional similarity assessments. This approach provides a more comprehensive evaluation of disparities between LLMs and human experts, while offering valuable insights into specific areas where LLMs may have room for improvement. For example, using ELF-Gym we empirically demonstrate that, in the best-case scenario, LLMs can semantically capture approximately 56% of the golden features, but at the more demanding implementation level this overlap drops to 13%. Moreover, in other cases LLMs may fail completely, particularly on datasets that require complex features, indicating broad potential pathways for improvement.

The angular synchronization problem aims to accurately estimate (up to a constant additive phase) a set of unknown angles $\theta_1, \dots, \theta_n\in[0, 2\pi)$ from $m$ noisy measurements of their offsets $\theta_i-\theta_j$ mod $2\pi.$ Applications include, for example, sensor network localization, phase retrieval, and distributed clock synchronization. An extension of the problem to the heterogeneous setting (dubbed $k$-synchronization) is to estimate $k$ groups of angles simultaneously, given noisy observations (with unknown group assignment) from each group. Existing methods for angular synchronization usually perform poorly in high-noise regimes, which are common in applications. In this paper, we leverage neural networks for the angular synchronization problem, and its heterogeneous extension, by proposing GNNSync, a theoretically-grounded end-to-end trainable framework using directed graph neural networks. In addition, new loss functions are devised to encode synchronization objectives. Experimental results on extensive data sets demonstrate that GNNSync attains competitive, and often superior, performance against a comprehensive set of baselines for the angular synchronization problem and its extension, validating the robustness of GNNSync even at high noise levels.

A long-standing goal in deep learning has been to characterize the learning behavior of black-box models in a more interpretable manner. For graph neural networks (GNNs), considerable advances have been made in formalizing what functions they can represent, but whether GNNs will learn desired functions during the optimization process remains less clear. To fill this gap, we study their training dynamics in function space. In particular, we find that the optimization of GNNs through gradient descent implicitly leverages the graph structure to update the learned function. This phenomenon is dubbed as kernel-graph alignment, which has been empirically and theoretically corroborated. This new analytical framework from the optimization perspective enables interpretable explanations of when and why the learned GNN functions generalize, which are relevant to their limitations on heterophilic graphs. From a practical standpoint, it also provides high-level principles for designing new algorithms. We exemplify this by showing that a simple and efficient non-parametric algorithm, obtained by explicitly using graph structure to update the learned function, can consistently compete with nonlinear GNNs.

While graph machine learning, and notably graph neural networks (GNNs), have gained immense traction in recent years, application is predicated on access to a known input graph upon which predictive models can be trained. And indeed, within the most widely-studied public evaluation benchmarks such graphs are provided, with performance comparisons conditioned on curated data explicitly adhering to this graph. However, in real-world industrial applications, the situation is often quite different. Instead of a known graph, data are originally collected and stored across multiple tables in a repository, at times with ambiguous or incomplete relational structure. As such, to leverage the latest GNN architectures it is then up to a skilled data scientist to first manually construct a graph using intuition and domain knowledge, a laborious process that may discourage adoption in the first place. To narrow this gap and broaden the applicability of graph ML, we survey existing tools and strategies that can be combined to address the more fundamental problem of predictive tabular modeling over data native to multiple tables, with no explicit relational structure assumed a priori. This involves tracing a comprehensive path through related table join discovery and fuzzy table joining, column alignment, automated relational database (RDB) construction, extracting graphs from RDBs, graph sampling, and finally, graph-centric trainable predictive architectures. Although efforts to build deployable systems that integrate all of these components while minimizing manual effort remain in their infancy, this survey will nonetheless reduce barriers to entry and help steer the graph ML community towards promising research directions and wider real-world impact.

Given a relational database (RDB), how can we predict missing column values in some target table of interest? Although RDBs store vast amounts of rich, informative data spread across interconnected tables, the progress of predictive machine learning models as applied to such tasks arguably falls well behind advances in other domains such as computer vision or natural language processing. This deficit stems, at least in part, from the lack of established/public RDB benchmarks as needed for training and evaluation purposes. As a result, related model development thus far often defaults to tabular approaches trained on ubiquitous single-table benchmarks, or on the relational side, graph-based alternatives such as GNNs applied to a completely different set of graph datasets devoid of tabular characteristics. To more precisely target RDBs lying at the nexus of these two complementary regimes, we explore a broad class of baseline models predicated on: (i) converting multi-table datasets into graphs using various strategies equipped with efficient subsampling, while preserving tabular characteristics; and (ii) trainable models with well-matched inductive biases that output predictions based on these input subgraphs. Then, to address the dearth of suitable public benchmarks and reduce siloed comparisons, we assemble a diverse collection of (i) large-scale RDB datasets and (ii) coincident predictive tasks. From a delivery standpoint, we operationalize the above four dimensions (4D) of exploration within a unified, scalable open-source toolbox called 4DBInfer; please see https://github.com/awslabs/multi-table-benchmark .

A key performance bottleneck when training graph neural network (GNN) models on large, real-world graphs is loading node features onto a GPU. Due to limited GPU memory, expensive data movement is necessary to facilitate the storage of these features on alternative devices with slower access (e.g. CPU memory). Moreover, the irregularity of graph structures contributes to poor data locality which further exacerbates the problem. Consequently, existing frameworks capable of efficiently training large GNN models usually incur a significant accuracy degradation because of the currently-available shortcuts involved. To address these limitations, we instead propose FreshGNN, a general-purpose GNN mini-batch training framework that leverages a historical cache for storing and reusing GNN node embeddings instead of re-computing them through fetching raw features at every iteration. Critical to its success, the corresponding cache policy is designed, using a combination of gradient-based and staleness criteria, to selectively screen those embeddings which are relatively stable and can be cached, from those that need to be re-computed to reduce estimation errors and subsequent downstream accuracy loss. When paired with complementary system enhancements to support this selective historical cache, FreshGNN is able to accelerate the training speed on large graph datasets such as ogbn-papers100M and MAG240M by 3.4× up to 20.5× and reduce the memory access by 59%, with less than 1% influence on test accuracy.

Graph Neural Networks (GNNs) have displayed considerable promise in graph representation learning across various applications. The core learning process requires the initialization of model weight matrices within each GNN layer, which is typically accomplished via classic initialization methods such as Xavier initialization. However, these methods were originally motivated to stabilize the variance of hidden embeddings and gradients across layers of Feedforward Neural Networks (FNNs) and Convolutional Neural Networks (CNNs) to avoid vanishing gradients and maintain steady information flow. In contrast, within the GNN context classical initializations disregard the impact of the input graph structure and message passing on variance. In this paper, we analyze the variance of forward and backward propagation across GNN layers and show that the variance instability of GNN initializations comes from the combined effect of the activation function, hidden dimension, graph structure and message passing. To better account for these influence factors, we propose a new initialization method for Variance Instability Reduction within GNN Optimization (Virgo), which naturally tends to equate forward and backward variances across successive layers. We conduct comprehensive experiments on 15 datasets to show that Virgo can lead to superior model performance and more stable variance at initialization on node classification, link prediction and graph classification tasks.

Hypergraphs are a powerful abstraction for representing higher-order interactions between entities of interest. To exploit these relationships in making downstream predictions, a variety of hypergraph neural network architectures have recently been proposed, in large part building upon precursors from the more traditional graph neural network (GNN) literature. Somewhat differently, in this paper we begin by presenting an expressive family of parameterized, hypergraph-regularized energy functions. We then demonstrate how minimizers of these energies effectively serve as node embeddings that, when paired with a parameterized classifier, can be trained end-to-end via a supervised bilevel optimization process. Later, we draw parallels between the implicit architecture of the predictive models emerging from the proposed bilevel hypergraph optimization, and existing GNN architectures in common use. Empirically, we demonstrate state-of-the-art results on various hypergraph node classification benchmarks. Code is available at https://github.com/yxzwang/PhenomNN.

Although the variational autoencoder (VAE) represents a widely-used deep generative model, the underlying energy function when applied to continuous data remains poorly understood. In fact, most prior theoretical analysis has assumed a simplified affine decoder such that the model collapses to probabilistic PCA, a restricted regime whereby existing classical algorithms can also be trivially applied to guarantee globally optimal solutions. To push our understanding into more complex, practically-relevant settings, this paper instead adopts a deceptively sophisticated single-layer decoder that nonetheless allows the VAE to address the fundamental challenge of learning optimally sparse representations of continuous data originating from popular multiple-response regression models. In doing so, we can then examine VAE properties within the non-trivial context of solving difficult, NP-hard inverse problems. More specifically, we prove rigorous conditions which guarantee that any minimum of the VAE energy (local or global) will produce the optimally sparse latent representation, meaning zero reconstruction error using a minimal number of active latent dimensions. This is ultimately possible because VAE marginalization over the latent posterior selectively smooths away bad local minima as has been conjectured but not actually proven in prior work. We then discuss how equivalent-capacity deterministic autoencoders, even with appropriate sparsity-promoting regularization of the latent space, maintain bad local minima that do not correspond with such parsimonious representations. Overall, these results serve to elucidate key properties of the VAE loss surface relative to finding low-dimensional structure in data.

We present a graph variational autoencoder with a structured prior for generating the layout of indoor 3D scenes. Given the room type (e.g., living room or library) and the room layout (e.g., room elements such as floor and walls), our architecture generates a collection of objects (e.g., furniture items such as sofa, table and chairs) that is consistent with the room type and layout. This is a challenging problem because the generated scene needs to satisfy multiple constrains, e.g., each object should lie inside the room and two objects should not occupy the same volume. To address these challenges, we propose a deep generative model that encodes these relationships as soft constraints on an attributed graph (e.g., the nodes capture attributes of room and furniture elements, such as shape, class, pose and size, and the edges capture geometric relationships such as relative orientation). The architecture consists of a graph encoder that maps the input graph to a structured latent space, and a graph decoder that generates a furniture graph, given a latent code and the room graph. The latent space is modeled with autoregressive priors, which facilitates the generation of highly structured scenes. We also propose an efficient training procedure that combines matching and constrained learning. Experiments on the 3D-FRONT dataset show that our method produces scenes that are diverse and are adapted to the room layout.

Decoding brain activity from non-invasive electroencephalography (EEG) is crucial for brain-computer interfaces (BCIs) and the study of brain disorders. Notably, end-to-end EEG decoding has gained widespread popularity in recent years owing to the remarkable advances in deep learning research. However, many EEG studies suffer from limited sample sizes, making it difficult for existing deep learning models to effectively generalize to highly noisy EEG data. To address this fundamental limitation, this paper proposes a novel end-to-end EEG decoding algorithm that utilizes a low-rank weight matrix to encode both spatio-temporal filters and the classifier, all optimized under a principled sparse Bayesian learning (SBL) framework. Importantly, this SBL framework also enables us to learn hyperparameters that optimally penalize the model in a Bayesian fashion. The proposed decoding algorithm is systematically benchmarked on five motor imagery BCI EEG datasets ( N=192) and an emotion recognition EEG dataset ( N=45), in comparison with several contemporary algorithms, including end-to-end deep-learning-based EEG decoding algorithms. The classification results demonstrate that our algorithm significantly outperforms the competing algorithms while yielding neurophysiologically meaningful spatio-temporal patterns. Our algorithm therefore advances the state-of-the-art by providing a novel EEG-tailored machine learning tool for decoding brain activity.

Recovering global rankings from pairwise comparisons has wide applications from time synchronization to sports team ranking. Pairwise comparisons corresponding to matches in a competition can be construed as edges in a directed graph (digraph), whose nodes represent e.g. competitors with an unknown rank. In this paper, we introduce neural networks into the ranking recovery problem by proposing the so-called GNNRank, a trainable GNN-based framework with digraph embedding. Moreover, new objectives are devised to encode ranking upsets/violations. The framework involves a ranking score estimation approach, and adds an inductive bias by unfolding the Fiedler vector computation of the graph constructed from a learnable similarity matrix. Experimental results on extensive data sets show that our methods attain competitive and often superior performance against baselines, as well as showing promising transfer ability. Codes and preprocessed data are at: \url{https://github.com/SherylHYX/GNNRank}.

Heterogeneous graph neural networks (GNNs) achieve strong performance on node classification tasks in a semi-supervised learning setting. However, as in the simpler homogeneous GNN case, message-passing-based heterogeneous GNNs may struggle to balance between resisting the oversmoothing that may occur in deep models, and capturing long-range dependencies of graph structured data. Moreover, the complexity of this trade-off is compounded in the heterogeneous graph case due to the disparate heterophily relationships between nodes of different types. To address these issues, we propose a novel heterogeneous GNN architecture in which layers are derived from optimization steps that descend a novel relation-aware energy function. The corresponding minimizer is fully differentiable with respect to the energy function parameters, such that bilevel optimization can be applied to effectively learn a functional form whose minimum provides optimal node representations for subsequent classification tasks. In particular, this methodology allows us to model diverse heterophily relationships between different node types while avoiding oversmoothing effects. Experimental results on 8 heterogeneous graph benchmarks demonstrates that our proposed method can achieve competitive node classification accuracy.

Prediction over tabular data is an essential and fundamental problem in many important downstream tasks. However, existing methods either take a data instance of the table independently as input or do not fully utilize the multi-row features and labels to directly change and enhance the target data representations. In this paper, we propose to 1) construct a hypergraph from relevant data instance retrieval to model the cross-row and cross-column patterns of those instances, and 2) perform message Propagation to Enhance the target data instance representation for Tabular prediction tasks. Specifically, our specially-designed message propagation step benefits from 1) the fusion of label and features during propagation, and 2) locality-aware multiplicative high-order interaction between features. Experiments on two important tabular prediction tasks validate the superiority of the proposed PET model against other baselines. Additionally, we demonstrate the effectiveness of the model components and the feature enhancement ability of PET via various ablation studies and visualizations. The code is available at https://github.com/KounianhuaDu/PET.

Graph neural networks have been extensively studied for learning with inter-connected data. Despite this, recent evidence has revealed GNNs' deficiencies related to over-squashing, heterophily, handling long-range dependencies, edge incompleteness and particularly, the absence of graphs altogether. While a plausible solution is to learn new adaptive topology for message passing, issues concerning quadratic complexity hinder simultaneous guarantees for scalability and precision in large networks. In this paper, we introduce a novel all-pair message passing scheme for efficiently propagating node signals between arbitrary nodes, as an important building block for a new class of Transformer networks for node classification on large graphs, dubbed as NodeFormer. Specifically, the efficient computation is enabled by a kernerlized Gumbel-Softmax operator that reduces the algorithmic complexity to linearity w.r.t. node numbers for learning latent graph structures from large, potentially fully-connected graphs in a differentiable manner. We also provide accompanying theory as justification for our design. Extensive experiments demonstrate the promising efficacy of the method in various tasks including node classification on graphs (with up to 2M nodes) and graph-enhanced applications (e.g., image classification) where input graphs are missing. The codes are available at https://github.com/qitianwu/NodeFormer.

Deep learning models such as the Transformer are often constructed by heuristics and experience. To provide a complementary foundation, in this work we study the following problem: Is it possible to find an energy function underlying the Transformer model, such that descent steps along this energy correspond with the Transformer forward pass? By finding such a function, we can reinterpret Transformers as the unfolding of an interpretable optimization process. This unfolding perspective has been frequently adopted in the past to elucidate more straightforward deep models such as MLPs and CNNs; however, it has thus far remained elusive obtaining a similar equivalence for more complex models with self-attention mechanisms like the Transformer. To this end, we first outline several major obstacles before providing companion techniques to at least partially address them, demonstrating for the first time a close association between energy function minimization and deep layers with self-attention. This interpretation contributes to our intuition and understanding of Transformers, while potentially laying the ground-work for new model designs.

Amodal perception requires inferring the full shape of an object that is partially occluded. This task is particularly challenging on two levels: (1) it requires more information than what is contained in the instant retina or imaging sensor, (2) it is difficult to obtain enough well-annotated amodal labels for supervision. To this end, this paper develops a new framework of Self-supervised amodal Video object segmentation (SaVos). Our method efficiently leverages the visual information of video temporal sequences to infer the amodal mask of objects. The key intuition is that the occluded part of an object can be explained away if that part is visible in other frames, possibly deformed as long as the deformation can be reasonably learned. Accordingly, we derive a novel self-supervised learning paradigm that efficiently utilizes the visible object parts as the supervision to guide the training on videos. In addition to learning type prior to complete masks for known types, SaVos also learns the spatiotemporal prior, which is also useful for the amodal task and could generalize to unseen types. The proposed framework achieves the state-of-the-art performance on the synthetic amodal segmentation benchmark FISHBOWL and the real world benchmark KINS-Video-Car. Further, it lends itself well to being transferred to novel distributions using test-time adaptation, outperforming existing models even after the transfer to a new distribution.

Although the variational autoencoder (VAE) and its conditional extension (CVAE) are capable of state-of-the-art results across multiple domains, their precise behavior is still not fully understood, particularly in the context of data (like images) that lie on or near a low-dimensional manifold. For example, while prior work has suggested that the globally optimal VAE solution can learn the correct manifold dimension, a necessary (but not sufficient) condition for producing samples from the true data distribution, this has never been rigorously proven. Moreover, it remains unclear how such considerations would change when various types of conditioning variables are introduced, or when the data support is extended to a union of manifolds (e.g., as is likely the case for MNIST digits and related). In this work, we address these points by first proving that VAE global minima are indeed capable of recovering the correct manifold dimension. We then extend this result to more general CVAEs, demonstrating practical scenarios whereby the conditioning variables allow the model to adaptively learn manifolds of varying dimension across samples. Our analyses, which have practical implications for various CVAE design choices, are also supported by numerical results on both synthetic and real-world datasets.

State-of-the-art face restoration methods employ deep convolutional neural networks (CNNs) to learn a mapping between degraded and sharp facial patterns by exploring local appearance knowledge. However, most of these methods do not well exploit facial structures and identity information, and only deal with task-specific face restoration (e.g., face super-resolution or deblurring). In this paper, we propose cross-tasks and cross-models plug-and-play 3D facial priors to explicitly embed the network with the sharp facial structures for general face restoration tasks. Our 3D priors are the first to explore 3D morphable knowledge based on the fusion of parametric descriptions of face attributes (e.g., identity, facial expression, texture, illumination, and face pose). Furthermore, the priors can easily be incorporated into any network and are very efficient in improving the performance and accelerating the convergence speed. Firstly, a 3D face rendering branch is set up to obtain 3D priors of salient facial structures and identity knowledge. Secondly, for better exploiting this hierarchical information (i.e., intensity similarity, 3D facial structure, and identity content), a spatial attention module is designed for the image restoration problems. Extensive face restoration experiments including face super-resolution and deblurring demonstrate that the proposed 3D priors achieve superior face restoration results over the state-of-the-art algorithms.

Cycle-consistent training is widely used for jointly learning a forward and inverse mapping between two domains of interest without the cumbersome requirement of collecting matched pairs within each domain. In this regard, the implicit assumption is that there exists (at least approximately) a ground-truth bijection such that a given input from either domain can be accurately reconstructed from successive application of the respective mappings. But in many applications no such bijection can be expected to exist and large reconstruction errors can compromise the success of cycle-consistent training. As one important instance of this limitation, we consider practically-relevant situations where there exists a many-to-one or surjective mapping between domains. To address this regime, we develop a conditional variational autoencoder (CVAE) approach that can be viewed as converting surjective mappings to implicit bijections whereby reconstruction errors in both directions can be minimized, and as a natural byproduct, realistic output diversity can be obtained in the one-to-many direction. As theoretical motivation, we analyze a simplified scenario whereby minima of the proposed CVAE-based energy function align with the recovery of ground-truth surjective mappings. On the empirical side, we consider a synthetic image dataset with known ground-truth, as well as a real-world application involving natural language generation from knowledge graphs and vice versa, a prototypical surjective case. For the latter, our CVAE pipeline can capture such many-to-one mappings during cycle training while promoting textural diversity for graph-to-text tasks.

Three-dimensional scanning by means of structured light illumination is an active imaging technique involving projecting and capturing a series of striped patterns and then using the observed warping of stripes to reconstruct the target object’s surface through triangulating each pixel in the camera to a unique projector coordinate corresponding to a particular feature in the projected patterns. The undesirable phenomenon of multi-path occurs when a camera pixel simultaneously sees features from multiple projector coordinates. Bimodal multi-path is a particularly common situation found along step edges, where the camera pixel sees both a foreground and background surface. Generalized from bimodal multi-path, this paper examines the phenomenon of sparse or N-modal multi-path as a more general case, where the camera pixel sees no fewer than two reflective surfaces, resulting in decoding errors. Using fringe projection profilometry, our proposed solution is to treat each camera pixel as an underdetermined linear system of equations and to find the sparsest (least number of paths) solution by taking an application-specific Bayesian learning approach. We validate this algorithm with both simulations and a number of challenging real-world scenarios, demonstrating that it outperforms state-of-the-art techniques.

Linear inverse problems appear in many applications, where different algorithms are typically employed to solve each inverse problem. Nowadays, the rapid development of deep learning (DL) provides a fresh perspective for solving the linear inverse problem: a number of well-designed network architectures results in state-of-the-art performance in many applications. In this overview paper, we present the combination of the DL and the Plug-and-Play priors (PPP) framework, showcasing how it allows solving various inverse problems by leveraging the impressive capabilities of existing DL based denoising algorithms. Open challenges and potential future directions along this line of research are also discussed.

We propose a hierarchical graph neural network (GNN) model that learns how to cluster a set of images into an unknown number of identities using a training set of images annotated with labels belonging to a disjoint set of identities. Our hierarchical GNN uses a novel approach to merge connected components predicted at each level of the hierarchy to form a new graph at the next level. Unlike fully unsupervised hierarchical clustering, the choice of grouping and complexity criteria stems naturally from supervision in the training set. The resulting method, Hi-LANDER, achieves an average of 49% improvement in F-score and 7% increase in Normalized Mutual Information (NMI) relative to current GNN-based clustering algorithms. Additionally, state-of-the-art GNN-based methods rely on separate models to predict linkage probabilities and node densities as intermediate steps of the clustering process. In contrast, our unified framework achieves a three-fold decrease in computational cost. Our training and inference code are released1.

Despite the recent success of graph neural networks (GNN), common architectures often exhibit significant limitations, including sensitivity to oversmoothing, long-range dependencies, and spurious edges, e.g., as can occur as a result of graph heterophily or adversarial attacks. To at least partially address these issues within a simple transparent framework, we consider a new family of GNN layers designed to mimic and integrate the update rules of two classical iterative algorithms, namely, proximal gradient descent and iterative reweighted least squares (IRLS). The former defines an extensible base GNN architecture that is immune to oversmoothing while nonetheless capturing long-range dependencies by allowing arbitrary propagation steps. In contrast, the latter produces a novel attention mechanism that is explicitly anchored to an underlying end-to-end energy function, contributing stability with respect to edge uncertainty. When combined we obtain an extremely simple yet robust model that we evaluate across disparate scenarios including standardized benchmarks, adversarially-perturbated graphs, graphs with heterophily, and graphs involving long-range dependencies. In doing so, we compare against SOTA GNN approaches that have been explicitly designed for the respective task, achieving competitive or superior node classification accuracy. Our code is available at https://github.com/FFTYYY/TWIRLS. And for an extended version of this work, please see https://arxiv.org/abs/2103.06064.

A number of recent studies of continuous variational autoencoder (VAE) models have noted, either directly or indirectly, the tendency of various parameter gradients to drift towards infinity during training. Because such gradients could potentially contribute to numerical instabilities, and are often framed as a problematic phenomena to be avoided, it may be tempting to shift to alternative energy functions that guarantee bounded gradients. But it remains an open question: What might the unintended consequences of such a restriction be? To address this issue, we examine how unbounded gradients relate to the regularization of a broad class of autoencoder-based architectures, including VAE models, as applied to data lying on or near a low-dimensional manifold (e.g., natural images). Our main finding is that, if the ultimate goal is to simultaneously avoid over-regularization (high reconstruction errors, sometimes referred to as posterior collapse) and under-regularization (excessive latent dimensions are not pruned from the model), then an autoencoder-based energy function with infinite gradients around optimal representations is provably required per a certain technical sense which we carefully detail. Given that both over- and under-regularization can directly lead to poor generated sample quality or suboptimal feature selection, this result suggests that heuristic modifications to or constraints on the VAE energy function may at times be ill-advised, and large gradients should be accommodated to the extent possible.

Learning the distribution of future trajectories conditioned on the past is a crucial problem for understanding multi-agent systems. This is challenging because humans make decisions based on complex social relations and personal intents, resulting in highly complex uncertainties over trajectories. To address this problem, we propose a conditional deep generative model that combines advances in graph neural networks. The prior and recognition model encodes two types of latent codes for each agent: an inter-agent latent code to represent social relations and an intra-agent latent code to represent agent intentions. The decoder is carefully devised to leverage the codes in a disentangled way to predict multi-modal future trajectory distribution. Specifically, a graph attention network built upon inter-agent latent code is used to learn continuous pair-wise relations, and an agent's motion is controlled by its latent intents and its observations of all other agents. Through experiments on both synthetic and real-world datasets, we show that our model outperforms previous work in multiple performance metrics. We also show that our model generates realistic multi-modal trajectories.

Graph neural networks (GNN) have recently emerged as a vehicle for applying deep network architectures to graph and relational data. However, given the increasing size of industrial datasets, in many practical situations, the message passing computations required for sharing information across GNN layers are no longer scalable. Although various sampling methods have been introduced to approximate full-graph training within a tractable budget, there remain unresolved complications such as high variances and limited theoretical guarantees. To address these issues, we build upon existing work and treat GNN neighbor sampling as a multi-armed bandit problem but with a newly-designed reward function that introduces some degree of bias designed to reduce variance and avoid unstable, possibly-unbounded pay outs. And unlike prior bandit-GNN use cases, the resulting policy leads to near-optimal regret while accounting for the GNN training dynamics introduced by SGD. From a practical standpoint, this translates into lower variance estimates and competitive or superior test accuracy across several benchmarks.

We introduce a conceptually simple yet effective model for self-supervised representation learning with graph data. It follows the previous methods that generate two views of an input graph through data augmentation. However, unlike contrastive methods that focus on instance-level discrimination, we optimize an innovative feature-level objective inspired by classical Canonical Correlation Analysis. Compared with other works, our approach requires none of the parameterized mutual information estimator, additional projector, asymmetric structures, and most importantly, negative samples which can be costly. We show that the new objective essentially 1) aims at discarding augmentation-variant information by learning invariant representations, and 2) can prevent degenerated solutions by decorrelating features in different dimensions. Our theoretical analysis further provides an understanding for the new objective which can be equivalently seen as an instantiation of the Information Bottleneck Principle under the self-supervised setting. Despite its simplicity, our method performs competitively on seven public graph datasets.

In narrow asymptotic settings Gaussian VAE models of continuous data have been shown to possess global optima aligned with ground-truth distributions. Even so, it is well known that poor solutions whereby the latent posterior collapses to an uninformative prior are sometimes obtained in practice. However, contrary to conventional wisdom that largely assigns blame for this phenomena on the undue influence of KL-divergence regularization, we will argue that posterior collapse is, at least in part, a direct consequence of bad local minima inherent to the loss surface of deep autoencoder networks. In particular, we prove that even small nonlinear perturbations of affine VAE decoder models can produce such minima, and in deeper models, analogous minima can force the VAE to behave like an aggressive truncation operator, provably discarding information along all latent dimensions in certain circumstances. Regardless, the underlying message here is not meant to undercut valuable existing explanations of posterior collapse, but rather, to refine the discussion and elucidate alternative risk factors that may have been previously underappreciated.

The recent, counter-intuitive discovery that deep generative models (DGMs) can frequently assign a higher likelihood to outliers has implications for both outlier detection applications as well as our overall understanding of generative modeling. In this work, we present a possible explanation for this phenomenon, starting from the observation that a model's typical set and high-density region may not conincide. From this vantage point we propose a novel outlier test, the empirical success of which suggests that the failure of existing likelihood-based outlier tests does not necessarily imply that the corresponding generative model is uncalibrated. We also conduct additional experiments to help disentangle the impact of low-level texture versus high-level semantics in differentiating outliers. In aggregate, these results suggest that modifications to the standard evaluation practices and benchmarks commonly applied in the literature are needed.

Removing undesirable reflections from a single image captured through a glass window is of practical importance to visual computing systems. Although state-of-the-art methods can obtain decent results in certain situations, performance declines significantly when tackling more general real-world cases. These failures stem from the intrinsic difficulty of single image reflection removal -- the fundamental ill-posedness of the problem, and the insufficiency of densely-labeled training data needed for resolving this ambiguity within learning-based neural network pipelines. In this paper, we address these issues by exploiting targeted network enhancements and the novel use of misaligned data. For the former, we augment a baseline network architecture by embedding context encoding modules that are capable of leveraging high-level contextual clues to reduce indeterminacy within areas containing strong reflections. For the latter, we introduce an alignment-invariant loss function that facilitates exploiting misaligned real-world training data that is much easier to collect. Experimental results collectively show that our method outperforms the state-of-the-art with aligned data, and that significant improvements are possible when using additional misaligned data.

Existing face deblurring methods only consider single frames and do not account for facial structure and identity information. These methods struggle to deblur face videos that exhibit significant pose variations and misalignment. In this paper we propose a novel face video deblurring network capitalizing on 3D facial priors. The model consists of two main branches: i) a face video deblurring sub-network based on an encoder-decoder architecture, and ii) a 3D face reconstruction and rendering branch for predicting 3D priors of salient facial structures and identity knowledge. These structures encourage the deblurring branch to generate sharp faces with detailed structures. Our method not only uses low-level information (i.e., image intensity), but also middle-level information (i.e., 3D facial structure) and high-level knowledge (i.e., identity content) to further explore spatial constraints of facial components from blurry face frames. Extensive experimental results demonstrate that the proposed algorithm performs favorably against the state-of-the-art methods.

While invaluable for many computer vision applications, decomposing a natural image into intrinsic reflectance and shading layers represents a challenging, underdetermined inverse problem. As opposed to strict reliance on conventional optimization or filtering solutions with strong prior assumptions, deep learning based approaches have also been proposed to compute intrinsic image decompositions when granted access to sufficient labeled training data. The downside is that current data sources are quite limited, and broadly speaking fall into one of two categories: either dense fully-labeled images in synthetic/narrow settings, or weakly-labeled data from relatively diverse natural scenes. In contrast to many previous learning-based approaches, which are often tailored to the structure of a particular dataset (and may not work well on others), we adopt core network structures that universally reflect loose prior knowledge regarding the intrinsic image formation process and can be largely shared across datasets. We then apply flexibly supervised loss layers that are customized for each source of ground truth labels. The resulting deep architecture achieves state-of-the-art results on all of the major intrinsic image benchmarks, and runs considerably faster than most at test time.

Neural networks can be compressed to reduce memory and computational requirements, or to increase accuracy by facilitating the use of a larger base architecture. In this paper we focus on pruning individual neurons, which can simultaneously trim model size, FLOPs, and run-time memory. To improve upon the performance of existing compression algorithms we utilize the information bottleneck principle instantiated via a tractable variational bound. Minimization of this information theoretic bound reduces the redundancy between adjacent layers by aggregating useful information into a subset of neurons that can be preserved. In contrast, the activations of disposable neurons are shut off via an attractive form of sparse regularization that emerges naturally from this framework, providing tangible advantages over traditional sparsity penalties without contributing additional tuning parameters to the energy landscape. We demonstrate state-of-the-art compression rates across an array of datasets and network architectures.

Variational autoencoders (VAE) represent a popular, flexible form of deep generative model that can be stochastically fit to samples from a given random process using an information-theoretic variational bound on the true underlying distribution. Once so-obtained, the model can be putatively used to generate new samples from this distribution, or to provide a low-dimensional latent representation of existing samples. While quite effective in numerous application domains, certain important mechanisms which govern the behavior of the VAE are obfuscated by the intractable integrals and resulting stochastic approximations involved. Moreover, as a highly non-convex model, it remains unclear exactly how minima of the underlying energy relate to original design purposes. We attempt to better quantify these issues by analyzing a series of tractable special cases of increasing complexity. In doing so, we unveil interesting connections with more traditional dimensionality reduction models, as well as an intrinsic yet underappreciated propensity for robustly dismissing sparse outliers when estimating latent manifolds. With respect to the latter, we demonstrate that the VAE can be viewed as the natural evolution of recent robust PCA models, capable of learning nonlinear manifolds of unknown dimension obscured by gross corruptions.

Image smoothing represents a fundamental component of many disparate computer vision and graphics applications. In this paper, we present a unified unsupervised (label-free) learning framework that facilitates generating flexible and high-quality smoothing effects by directly learning from data using deep convolutional neural networks (CNNs). The heart of the design is the training signal as a novel energy function that includes an edge-preserving regularizer which helps maintain important yet potentially vulnerable image structures, and a spatially-adaptive L p flattening criterion which imposes different forms of regularization onto different image regions for better smoothing quality. We implement a diverse set of image smoothing solutions employing the unified framework targeting various applications such as, image abstraction, pencil sketching, detail enhancement, texture removal and content-aware image manipulation, and obtain results comparable with or better than previous methods. Moreover, our method is extremely fast with a modern GPU (e.g, 200 fps for 1280×720 images).

This paper proposes a deep neural network structure that exploits edge information in addressing representative low-level vision tasks such as layer separation and image filtering. Unlike most other deep learning strategies applied in this context, our approach tackles these challenging problems by estimating edges and reconstructing images using only cascaded convolutional layers arranged such that no handcrafted or application-specific image-processing components are required. We apply the resulting transferrable pipeline to two different problem domains that are both sensitive to edges, namely, single image reflection removal and image smoothing. For the former, using a mild reflection smoothness assumption and a novel synthetic data generation method that acts as a type of weak supervision, our network is able to solve much more difficult reflection cases that cannot be handled by previous methods. For the latter, we also exceed the state-of-the-art quantitative and qualitative results by wide margins. In all cases, the proposed framework is simple, fast, and easy to transfer across disparate domains.

The iterations of many first-order algorithms, when applied to minimizing common regularized regression functions, often resemble neural network layers with pre-specified weights. This observation has prompted the development of learning-based approaches that purport to replace these iterations with enhanced surrogates forged as DNN models from available training data. For example, important NP-hard sparse estimation problems have recently benefitted from this genre of upgrade, with simple feedforward or recurrent networks ousting proximal gradient-based iterations. Analogously, this paper demonstrates that more powerful Bayesian algorithms for promoting sparsity, which rely on complex multi-loop majorization-minimization techniques, mirror the structure of more sophisticated long short-term memory (LSTM) networks, or alternative gated feedback networks previously designed for sequence prediction. As part of this development, we examine the parallels between latent variable trajectories operating across multiple time-scales during optimization, and the activations within deep network structures designed to adaptively model such characteristic sequences. The resulting insights lead to a novel sparse estimation system that, when granted training data, can estimate optimal solutions efficiently in regimes where other algorithms fail, including practical direction-of-arrival (DOA) and 3D geometry recovery problems. The underlying principles we expose are also suggestive of a learning process for a richer class of multi-loop algorithms in other domains.

This paper revisits the classical Bayesian face recognition algorithm from Baback Moghaddam et al. and proposes enhancements tailored to face verification, the problem of predicting whether or not a pair of facial images share the same identity. Like a variety of face verification algorithms, the original Bayesian face model only considers the appearance difference between two faces rather than the raw images themselves. However, we argue that such a fixed and blind projection may prematurely reduce the separability between classes. Consequently, we model two facial images jointly with an appropriate prior that considers intra- and extra-personal variations over the image pairs. This joint formulation is trained using a principled EM algorithm, while testing involves only efficient closed-formed computations that are suitable for real-time practical deployment. Supporting theoretical analyses investigate computational complexity, scale-invariance properties, and convergence issues. We also detail important relationships with existing algorithms, such as probabilistic linear discriminant analysis and metric learning. Finally, on extensive experimental evaluations, the proposed model is superior to the classical Bayesian face algorithm and many alternative state-of-the-art supervised approaches, achieving the best test accuracy on three challenging datasets, Labeled Face in Wild, Multi-PIE, and YouTube Faces, all with unparalleled computational efficiency.

Variational Bayesian (VB) approximations anchor a wide variety of probabilistic models, where tractable posterior inference is almost never possible. Typically based on the so-called VB mean-field approximation to the Kullback-Leibler divergence, a posterior distribution is sought that factorizes across groups of latent variables such that, with the distributions of all but one group of variables held fixed, an optimal closed-form distribution can be obtained for the remaining group, with differing algorithms distinguished by how different variables are grouped and ultimately factored. This basic strategy is particularly attractive when estimating structured low-dimensional models of high-dimensional data, exemplified by the search for minimal rank and/or sparse approximations to observed data. To this end, VB models are frequently deployed across applications including multi-task learning, robust PCA, subspace clustering, matrix completion, affine rank minimization, source localization, compressive sensing, and assorted combinations thereof. Perhaps surprisingly however, there exists almost no attendant theoretical explanation for how various VB factorizations operate, and in which situations one may be preferable to another. We address this relative void by comparing arguably two of the most popular factorizations, one built upon Gaussian scale mixture priors, the other bilinear Gaussian priors, both of which can favor minimal rank or sparsity depending on the context. More specifically, by reexpressing the respective VB objective functions, we weigh multiple factors related to local minima avoidance, feature transformation invariance and correlation, and computational complexity to arrive at insightful conclusions useful in explaining performance and deciding which VB flavor is advantageous. We also envision that the principles explored here are quite relevant to other structured inverse problems where VB serves as a viable solution.

In this paper, we develop a new aggregation technique to reduce the cost of surveying. Our method aims to jointly estimate a vector of target quantities such as public opinion or voter intent across time and maintain good estimates when using only a fraction of the data. Inspired by the James-Stein estimator, we resolve this challenge by shrinking the estimates to a global mean which is assumed to have a sparse representation in some known basis. This assumption has lead to two different methods for estimating the global mean: orthogonal matching pursuit and deep learning. Both of which significantly reduce the number of samples needed to achieve good estimates of the true means of the data and, in the case of presidential elections, can estimate the outcome of the 2012 United States elections while saving hundreds of thousands of samples and maintaining accuracy.

Commonly used in many applications, robust PCA represents an algorithmic attempt to reduce the sensitivity of classical PCA to outliers. The basic idea is to learn a decomposition of some data matrix of interest into low rank and sparse components, the latter representing unwanted outliers. Although the resulting problem is typically NP-hard, convex relaxations provide a computationally-expedient alternative with theoretical support. However, in practical regimes performance guarantees break down and a variety of non-convex alternatives, including Bayesian-inspired models, have been proposed to boost estimation quality. Unfortunately though, without additional a priori knowledge none of these methods can significantly expand the critical operational range such that exact principal subspace recovery is possible. Into this mix we propose a novel pseudo-Bayesian algorithm that explicitly compensates for design weaknesses in many existing non-convex approaches leading to state-of-the-art performance with a sound analytical foundation.

The iterations of many sparse estimation algorithms are comprised of a fixed linear filter cascaded with a thresholding nonlinearity, which collectively resemble a typical neural network layer. Consequently, a lengthy sequence of algorithm iterations can be viewed as a deep network with shared, hand-crafted layer weights. It is therefore quite natural to examine the degree to which a learned network model might act as a viable surrogate for traditional sparse estimation in domains where ample training data is available. While the possibility of a reduced computational budget is readily apparent when a ceiling is imposed on the number of layers, our work primarily focuses on estimation accuracy. In particular, it is well-known that when a signal dictionary has coherent columns, as quantified by a large RIP constant, then most tractable iterative algorithms are unable to find maximally sparse representations. In contrast, we demonstrate both theoretically and empirically the potential for a trained deep network to recover minimal $\ell_0$-norm representations in regimes where existing methods fail. The resulting system, which can effectively learn novel iterative sparse estimation algorithms, is deployed on a practical photometric stereo estimation problem, where the goal is to remove sparse outliers that can disrupt the estimation of surface normals from a 3D scene.

We present a novel method for capturing real-world, spatially-varying surface reflectance from a small number of object views ( k ). Our key observation is that a specific target's reflectance can be represented by a small number of custom basis materials ( N ) convexly blended by an even smaller number of non-zero weights at each point ( n ). Based on this sparse basis/sparser blend model, we develop an SVBRDF reconstruction algorithm that jointly solves for n , N , the basis BRDFs, and their spatial blend weights with an alternating iterative optimization, each step of which solves a linearly-constrained quadratic programming problem. We develop a numerical tool that lets us estimate the number of views required and analyze the effect of lighting and geometry on reconstruction quality. We validate our method with images rendered from synthetic BRDFs, and demonstrate convincing results on real objects of pre-scanned shape and lit by uncontrolled natural illumination, from very few or even a single input image.

Linear discriminant analysis (LDA) represents a simple yet powerful technique for partitioning a p-dimensional feature vector into one of K classes based on a linear projection learned from N labeled observations. However, it is well-established that in the high-dimensional setting (p > N) the underlying projection estimator degenerates. Moreover, any linear discriminate function involving a large number of features may be difficult to interpret. To ameliorate these issues, two general categories of sparse LDA modifications have been proposed, both to reduce the number of active features and to stabilize the resulting projections. The first, based on optimal scoring, is more straightforward to implement and analyze but has been heavily criticized for its ambiguous connection with the original LDA formulation. In contrast, a second strategy applies sparse penalty functions directly to the original LDA objective but requires additional heuristic trade-off parameters, has unknown global and local minima properties, and requires a greedy sequential optimization procedure. In all cases the choice of sparse regularizer can be important, but no rigorous guidelines have been provided regarding which penalty might be preferable. Against this backdrop, we winnow down the broad space of candidate sparse LDA algorithms and promote a specific selection based on optimal scoring coupled with a particular, complementary sparse regularizer. This overall process ultimately progresses our understanding of sparse LDA in general, while leading to targeted modifications of existing algorithms that produce superior results in practice on three high-dimensional gene data sets.

The simultaneous sparse approximation problem is concerned with recovering a set of multichannel signals that share a common support pattern using incomplete or compressive measurements. Multichannel modifications of greedy algorithms like orthogonal matching pursuit (OMP), as well as convex mixed-norm extensions of the Lasso, have typically been deployed for efficient signal estimation. While accurate recovery is possible under certain circumstances, it has been established that these methods may all fail in regimes where traditional subspace techniques from array processing, notably the MUSIC algorithm, can provably succeed. Against this backdrop several recent hybrid algorithms have been developed that merge a subspace estimation step with OMP-like procedures to obtain superior results, sometimes with theoretical guarantees. In contrast, this paper considers a completely different approach built upon Bayesian compressive sensing. In particular, we demonstrate that minor modifications of standard Bayesian algorithms can naturally obtain the best of both worlds backed with theoretical and empirical support, surpassing the performance of existing hybrid MUSIC and convex simultaneous sparse approximation algorithms, especially when poor RIP conditions render alternative approaches ineffectual.

With the growing popularity of short-form video sharing platforms such as Instagram and Vine, there has been an increasing need for techniques that automatically extract highlights from video. Whereas prior works have approached this problem with heuristic rules or supervised learning, we present an unsupervised learning approach that takes advantage of the abundance of user-edited videos on social media websites such as YouTube. Based on the idea that the most significant sub-events within a video class are commonly present among edited videos while less interesting ones appear less frequently, we identify the significant sub-events via a robust recurrent auto-encoder trained on a collection of user-edited videos queried for each particular class of interest. The auto-encoder is trained using a proposed shrinking exponential loss function that makes it robust to noise in the web-crawled training data, and is configured with bidirectional long short term memory (LSTM) [5] cells to better model the temporal structure of highlight segments. Different from supervised techniques, our method can infer highlights using only a set of downloaded edited videos, without also needing their pre-edited counterparts which are rarely available online. Extensive experiments indicate the promise of our proposed solution in this challenging unsupervised setting.

Subspace segmentation is the process of clustering a set of data points that are assumed to lie on the union of multiple linear or affine subspaces, and is increasingly being recognized as a fundamental tool for data analysis in high dimensional settings. Arguably one of the most successful approaches is based on the observation that the sparsest representation of a given point with respect to a dictionary formed by the others involves nonzero coefficients associated with points originating in the same subspace. Such sparse representations are computed independently for each data point via \ell_1-norm minimization and then combined into an affinity matrix for use by a final spectral clustering step. The downside of this procedure is two-fold. First, unlike canonical compressive sensing scenarios with ideally-randomized dictionaries, the data-dependent dictionaries here are unavoidably highly structured, disrupting many of the favorable properties of the \ell_1 norm. Secondly, by treating each data point independently, we ignore useful relationships between points that can be leveraged for jointly computing such sparse representations. Consequently, we motivate a multi-task learning-based framework for learning coupled sparse representations leading to a segmentation pipeline that is both robust against correlation structure and tailored to generate an optimal affinity matrix. Theoretical analysis and empirical tests are provided to support these claims.

Many applications require recovering a matrix of minimal rank within an affine constraint set, with matrix completion a notable special case. Because the problem is NP-hard in general, it is common to replace the matrix rank with the nuclear norm, which acts as a convenient convex surrogate. While elegant theoretical conditions elucidate when this replacement is likely to be successful, they are highly restrictive and convex algorithms fail when the ambient rank is too high or when the constraint set is poorly structured. Non-convex alternatives fare somewhat better when carefully tuned; however, convergence to locally optimal solutions remains a continuing source of failure. Against this backdrop we derive a deceptively simple and parameter-free probabilistic PCA-like algorithm that is capable, over a wide battery of empirical tests, of successful recovery even at the theoretical limit where the number of measurements equals the degrees of freedom in the unknown low-rank matrix. Somewhat surprisingly, this is possible even when the affine constraint set is highly ill-conditioned. While proving general recovery guarantees remains evasive for non-convex algorithms, Bayesian-inspired or otherwise, we nonetheless show conditions whereby the underlying cost function has a unique stationary point located at the global optimum; no existing cost function we are aware of satisfies this property. The algorithm has also been successfully deployed on a computer vision application involving image rectification and a standard collaborative filtering benchmark.

We propose a novel dynamic reweighted ℓ2(DRℓ2) algorithm in the regime of dynamic compressive sensing. Our analysis shows that aiming to solve a Type II optimization problem, DRℓ2is effectively minimizing a `convex-concave' penalty in the coefficients that transitions from a convex region to a concave function using knowledge of past estimations. DRℓ2thus provides superior reconstruction performance compared with state-of-the-art dynamic CS algorithms.

Most conventional algorithms for non-Lambertian photometric stereo can be partitioned into two categories. The first category is built upon stable outlier rejection techniques while assuming a dense Lambertian structure for the inliers, and thus performance degrades when general diffuse regions are present. The second utilizes complex reflectance representations and non-linear optimization over pixels to handle non-Lambertian surfaces, but does not explicitly account for shadows or other forms of corrupting outliers. In this paper, we present a purely pixel-wise photometric stereo method that stably and efficiently handles various non-Lambertian effects by assuming that appearances can be decomposed into a sparse, non-diffuse component (e.g., shadows, specularities, etc.) and a diffuse component represented by a monotonic function of the surface normal and lighting dot-product. This function is constructed using a piecewise linear approximation to the inverse diffuse model, leading to closed-form estimates of the surface normals and model parameters in the absence of non-diffuse corruptions. The latter are modeled as latent variables embedded within a hierarchical Bayesian model such that we may accurately compute the unknown surface normals while simultaneously separating diffuse from non-diffuse components. Extensive evaluations are performed that show state-of-the-art performance using both synthetic and real-world images.

This paper describes a robust algorithm for estimating a single latent sharp image given multiple blurry and/or noisy observations. The underlying multi-image blind deconvolution problem is solved by linking all of the observations together via a Bayesian-inspired penalty function, which couples the unknown latent image along with a separate blur kernel and noise variance associated with each observation, all of which are estimated jointly from the data. This coupled penalty function enjoys a number of desirable properties, including a mechanism whereby the relative-concavity or sparsity is adapted as a function of the intrinsic quality of each corrupted observation. In this way, higher quality observations may automatically contribute more to the final estimate than heavily degraded ones, while troublesome local minima can largely be avoided. The resulting algorithm, which requires no essential tuning parameters, can recover a sharp image from a set of observations containing potentially both blurry and noisy examples, without knowing a priori the degradation type of each observation. Experimental results on both synthetic and real-world test images clearly demonstrate the efficacy of the proposed method.

This paper presents a robust algorithm for estimating a single latent sharp image given multiple blurry and/or noisy observations. The underlying multi-image blind deconvolution problem is solved by linking all of the observations together via a Bayesian-inspired penalty function which couples the unknown latent image, blur kernels, and noise levels together in a unique way. This coupled penalty function enjoys a number of desirable properties, including a mechanism whereby the relative-concavity or shape is adapted as a function of the intrinsic quality of each blurry observation. In this way, higher quality observations may automatically contribute more to the final estimate than heavily degraded ones. The resulting algorithm, which requires no essential tuning parameters, can recover a high quality image from a set of observations containing potentially both blurry and noisy examples, without knowing a priori the degradation type of each observation. Experimental results on both synthetic and real-world test images clearly demonstrate the efficacy of the proposed method.

Face verification involves determining whether a pair of facial images belongs to the same or different subjects. This problem can prove to be quite challenging in many important applications where labeled training data is scarce, e.g., family album photo organization software. Herein we propose a principled transfer learning approach for merging plentiful source-domain data with limited samples from some target domain of interest to create a classifier that ideally performs nearly as well as if rich target-domain data were present. Based upon a surprisingly simple generative Bayesian model, our approach combines a KL-divergence based regularizer/prior with a robust likelihood function leading to a scalable implementation via the EM algorithm. As justification for our design choices, we later use principles from convex analysis to recast our algorithm as an equivalent structured rank minimization problem leading to a number of interesting insights related to solution structure and feature-transform invariance. These insights help to both explain the effectiveness of our algorithm as well as elucidate a wide variety of related Bayesian approaches. Experimental testing with challenging datasets validate the utility of the proposed algorithm.

In this paper, we propose a simple but effective solution to the structured labeling problem: a fixed-point model. Recently, layered models with sequential classifiers/regressors have gained an increasing amount of interests for structural prediction. Here, we design an algorithm with a new perspective on layered models; we aim to find a fixed-point function with the structured labels being both the output and the input. Our approach alleviates the burden in learning multiple/different classifiers in different layers. We devise a training strategy for our method and provide justifications for the fixed-point function to be a contraction mapping. The learned function captures rich contextual information and is easy to train and test. On several widely used benchmark datasets, the proposed method observes significant improvement in both performance and efficiency over many state-of-the-art algorithms.

Typical blur from camera shake often deviates from the standard uniform convolutional assumption, in part because of problematic rotations which create greater blurring away from some unknown center point. Consequently, successful blind deconvolution for removing shake artifacts requires the estimation of a spatially-varying or non-uniform blur operator. Using ideas from Bayesian inference and convex analysis, this paper derives a non-uniform blind deblurring algorithm with several desirable, yet previously-unexplored attributes. The underlying objective function includes a spatially-adaptive penalty that couples the latent sharp image, non-uniform blur operator, and noise level together. This coupling allows the penalty to automatically adjust its shape based on the estimated degree of local blur and image structure such that regions with large blur or few prominent edges are discounted. Remaining regions with modest blur and revealing edges therefore dominate the overall estimation process without explicitly incorporating structure-selection heuristics. The algorithm can be implemented using an optimization strategy that is virtually parameter free and simpler than existing methods. Detailed theoretical analysis and empirical validation on real images serve to validate the proposed method.

This paper presents a robust photometric stereo method that effectively compensates for various non-Lambertian corruptions such as specularities, shadows, and image noise. We construct a constrained sparse regression problem that enforces both Lambertian, rank-3 structure and sparse, additive corruptions. A solution method is derived using a hierarchical Bayesian approximation to accurately estimate the surface normals while simultaneously separating the non-Lambertian corruptions. Extensive evaluations are performed that show state-of-the-art performance using both synthetic and real-world images.

For scene classification, patch-level linear features do not always work as well as handcrafted features. In this paper, we present a new model to greatly improve the usefulness of linear features in classification by introducing co-variance patterns. We analyze their properties, discuss the fundamental importance, and present a generative model to properly utilize them. With this set of covariance information, in our framework, even the most naive linear features that originally lack the vital ability in classification become powerful. Experiments show that the performance of our new covariance model based on linear features is comparable with or even better than handcrafted features in scene classification.

Sparse linear (or generalized linear) models combine a standard likelihood function with a sparse prior on the unknown coefficients. These priors can conveniently be expressed as a maximization over zero-mean Gaussians with different variance hyperparameters. Standard MAP estimation (Type I) involves maximizing over both the hyperparameters and coefficients, while an empirical Bayesian alternative (Type II) first marginalizes the coefficients and then maximizes over the hyperparameters, leading to a tractable posterior approximation. The underlying cost functions can be related via a dual-space framework from Wipf et al. (2011), which allows both the Type I or Type II objectives to be expressed in either coefficient or hyperparmeter space. This perspective is useful because some analyses or extensions are more conducive to development in one space or the other. Herein we consider the estimation of a trade-off parameter balancing sparsity and data fit. As this parameter is effectively a variance, natural estimators exist by assessing the problem in hyperparameter (variance) space, transitioning natural ideas from Type II to solve what is much less intuitive for Type I. In contrast, for analyses of update rules and sparsity properties of local and global solutions, as well as extensions to more general likelihood models, we can leverage coefficient-space techniques developed for Type I and apply them to Type II. For example, this allows us to prove that Type II-inspired techniques can be successful recovering sparse coefficients when unfavorable restricted isometry properties (RIP) lead to failure of popular L1 reconstructions. It also facilitates the analysis of Type II when non-Gaussian likelihood models lead to intractable integrations.

In the vast majority of recent work on sparse estimation algorithms, performance has been evaluated using ideal or quasi-ideal dictionaries (e.g., random Gaussian or Fourier) characterized by unit $\ell_2$ norm, incoherent columns or features. But in reality, these types of dictionaries represent only a subset of the dictionaries that are actually used in practice (largely restricted to idealized compressive sensing applications). In contrast, herein sparse estimation is considered in the context of structured dictionaries possibly exhibiting high coherence between arbitrary groups of columns and/or rows. Sparse penalized regression models are analyzed with the purpose of finding, to the extent possible, regimes of dictionary invariant performance. In particular, a Type II Bayesian estimator with a dictionary-dependent sparsity penalty is shown to have a number of desirable invariance properties leading to provable advantages over more conventional penalties such as the $\ell_1$ norm, especially in areas where existing theoretical recovery guarantees no longer hold. This can translate into improved performance in applications such as model selection with correlated features, source localization, and compressive sensing with constrained measurement directions.

Many practical methods for finding maximally sparse coefficient expansions involve solving a regression problem using a particular class of concave penalty functions. From a Bayesian perspective, this process is equivalent to maximum a posteriori (MAP) estimation using a sparsity-inducing prior distribution (Type I estimation). Using variational techniques, this distribution can always be conveniently expressed as a maximization over scaled Gaussian distributions modulated by a set of latent variables. Alternative Bayesian algorithms, which operate in latent variable space leveraging this variational representation, lead to sparse estimators reflecting posterior information beyond the mode (Type II estimation). Currently, it is unclear how the underlying cost functions of Type I and Type II relate, nor what relevant theoretical properties exist, especially with regard to Type II. Herein a common set of auxiliary functions is used to conveniently express both Type I and Type II cost functions in either coefficient or latent variable space facilitating direct comparisons. In coefficient space, the analysis reveals that Type II is exactly equivalent to performing standard MAP estimation using a particular class of dictionary- and noise-dependent, nonfactorial coefficient priors. One prior (at least) from this class maintains several desirable advantages over all possible Type I methods and utilizes a novel, nonconvex approximation to thel0norm with most, and in certain quantifiable conditions all, local minima smoothed away. Importantly, the global minimum is always left unaltered unlike standardl1-norm relaxations. This ensures that any appropriate descent method is guaranteed to locate the maximally sparse solution.

Finding maximally sparse representations from overcomplete feature dictionaries frequently involves minimizing a cost function composed of a likelihood (or data fit) term and a prior (or penalty function) that favors sparsity. While typically the prior is factorial, here we examine non-factorial alternatives that have a number of desirable properties relevant to sparse estimation and are easily implemented using an efficient, globally-convergent reweighted $\ell_1$ minimization procedure. The first method under consideration arises from the sparse Bayesian learning (SBL) framework. Although based on a highly non-convex underlying cost function, in the context of canonical sparse estimation problems, we prove uniform superiority of this method over the Lasso in that, (i) it can never do worse, and (ii) for any dictionary and sparsity profile, there will always exist cases where it does better. These results challenge the prevailing reliance on strictly convex penalty functions for finding sparse solutions. We then derive a new non-factorial variant with similar properties that exhibits further performance improvements in empirical tests. For both of these methods, as well as traditional factorial analogs, we demonstrate the effectiveness of reweighted $\ell_1$-norm algorithms in handling more general sparse estimation problems involving classification, group feature selection, and non-negativity constraints. As a byproduct of this development, a rigorous reformulation of sparse Bayesian classification (e.g., the relevance vector machine) is derived that, unlike the original, involves no approximation steps and descends a well-defined objective function.

The synchronous brain activity measured via MEG (or EEG) can be interpreted as arising from a collection (possibly large) of current dipoles or sources located throughout the cortex. Estimating the number, location, and orientation of these sources remains a challenging task, one that is significantly compounded by the effects of source correlations and the presence of interference from spontaneous brain activity, sensor noise, and other artifacts. This paper derives an empirical Bayesian method for addressing each of these issues in a principled fashion. The resulting algorithm guarantees descent of a cost function uniquely designed to handle unknown orientations and arbitrary correlations. Robust interference suppression is also easily incorporated. In a restricted setting, the proposed method is shown to have theoretically zero bias estimating both the location and orientation of multi-component dipoles even in the presence of correlations, unlike a variety of existing Bayesian localization methods or common signal processing techniques such as beamforming and sLORETA. Empirical results on both simulated and real data sets verify the efficacy of this approach.

A variety of Bayesian methods have recently been introduced for finding sparse representations from overcomplete dictionaries of candidate features. These methods often capitalize on latent structure inherent in sparse distributions to perform standard MAP estimation, variational Bayes, approximation using convex duality, or evidence maximization. Despite their reliance on sparsity-inducing priors however, these approaches may or may not actually lead to sparse representations in practice, and so it is a challenging task to determine which algorithm and sparse prior is appropriate. Rather than justifying prior selections and modelling assumptions based on the credibility of the full Bayesian model as is commonly done, this paper bases evaluations on the actual cost functions that emerge from each method. Two minimal conditions are postulated that ideally any sparse learning objective should satisfy. Out of all possible cost functions that can be obtained from the methods described above using (virtually) any sparse prior, a unique function is derived that satisfies these conditions. Both sparse Bayesian learning (SBL) and basis pursuit (BP) are special cases. Later, all methods are shown to be performing MAP estimation using potentially non-factorable implicit priors, which suggests new sparse learning cost functions.

Beamformers are spatial filters that pass source signals in particular focused locations while suppressing interference from elsewhere. The widely-used minimum variance adaptive beamformer (MVAB) creates such filters using a sample covariance estimate; however, the quality of this estimate deteriorates when the sources are correlated or the number of samples n is small. Herein, a modified beamformer is derived that replaces this problematic sample covariance with a robust maximum likelihood estimate obtained using the relevance vector machine (RVM), a Bayesian method for learning sparse models from possibly overcomplete feature sets. We prove that this substitution has the natural ability to remove the undesirable effects of correlations or limited data. When n becomes large and assuming uncorrelated sources, this method reduces to the exact MVAB. Simulations using direction-of-arrival data support these conclusions. Additionally, RVMs can potentially enhance a variety of traditional signal processing methods that rely on robust sample covariance estimates.

Automatic relevance determination (ARD), and the closely-related sparse Bayesian learning (SBL) framework, are effective tools for pruning large numbers of irrelevant features. However, popular update rules used for this process are either prohibitively slow in practice and/or heuristic in nature without proven convergence properties. This paper furnishes an alternative means of optimizing a general ARD cost function using an auxiliary function that can naturally be solved using a series of re-weighted L1 problems. The result is an efficient algorithm that can be implemented using standard convex programming toolboxes and is guaranteed to converge to a stationary point unlike existing methods. The analysis also leads to additional insights into the behavior of previous ARD updates as well as the ARD cost function. For example, the standard fixed-point updates of MacKay (1992) are shown to be iteratively solving a particular min-max problem, although they are not guaranteed to lead to a stationary point. The analysis also reveals that ARD is exactly equivalent to performing MAP estimation using a particular feature- and noise-dependent \textit{non-factorial} weight prior with several desirable properties over conventional priors with respect to feature selection. In particular, it provides a tighter approximation to the L0 quasi-norm sparsity measure than the L1 norm. Overall these results suggests alternative cost functions and update procedures for selecting features and promoting sparse solutions.

In this paper, we demonstrate a driver intent inference system that is based on lane positional information, vehicle parameters, and driver head motion. We present robust computer vision methods for identifying and tracking freeway lanes and driver head motion. These algorithms are then applied and evaluated on real-world data that are collected in a modular intelligent vehicle test bed. Analysis of the data for lane change intent is performed using a sparse Bayesian learning methodology. Finally, the system as a whole is evaluated using a novel metric and real-world data of vehicle parameters, lane position, and driver head motion.

The ill-posed nature of the MEG/EEG source localization problem requires the incorporation of prior assumptions when choosing an appropriate solution out of an infinite set of candidates. Bayesian methods are useful in this capacity because they allow these assumptions to be explicitly quantified. Recently, a number of empirical Bayesian approaches have been proposed that attempt a form of model selection by using the data to guide the search for an appropriate prior. While seemingly quite different in many respects, we apply a unifying framework based on automatic relevance determination (ARD) that elucidates various attributes of these methods and suggests directions for improvement. We also derive theoretical properties of this methodology related to convergence, local minima, and localization bias and explore connections with established algorithms.

We consider criteria for variational representations of non-Gaussian latent variables, and derive variational EM algorithms in general form. We establish a general equivalence among convex bounding methods, evidence based methods, and ensemble learning/Variational Bayes methods, which has previously been demonstrated only for particular cases.

Given a redundant dictionary of basis vectors (or atoms), our goal is to find maximally sparse representations of signals. Previously, we have argued that a sparse Bayesian learning (SBL) framework is particularly well-suited for this task, showing that it has far fewer local minima than other Bayesian-inspired strategies. In this paper, we provide further evi- dence for this claim by proving a restricted equivalence condition, based on the distribution of the nonzero generating model weights, whereby the SBL solution will equal the maximally sparse representation. We also prove that if these nonzero weights are drawn from an approximate Jef- freys prior, then with probability approaching one, our equivalence con- dition is satisfied. Finally, we motivate the worst-case scenario for SBL and demonstrate that it is still better than the most widely used sparse rep- resentation algorithms. These include Basis Pursuit (BP), which is based on a convex relaxation of the ℓ0 (quasi)-norm, and Orthogonal Match- ing Pursuit (OMP), a simple greedy strategy that iteratively selects basis vectors most aligned with the current residual.

Finding sparse representations of signals is an important problem in many application domains. Unfortunately, when the signal dictionary is overcomplete, finding the sparsest representation is NP-hard without some prior knowledge of the solution. However, suppose that we have access to such information. Is it possible to demonstrate any performance bounds in this restricted setting? We examine this question with respect to algorithms that minimize general /spl lscr//sub p/-norm-like diversity measures. Using randomized dictionaries, we analyze performance probabilistically under two conditions. First, when 0/spl les/p/spl les/1, we quantify (almost surely) the number and quality of every local minimum. Next, for the p=1 case, we extend the deterministic results of D.L. Donoho and M. Elad (see Proc. Nat. Acad. Sci., vol.100, no.5, 2003) by deriving explicit confidence intervals for a theoretical equivalence bound, under which the minimum /spl lscr//sub 1/-norm solution is guaranteed to equal the maximally sparse solution. These results elucidate our previous empirical studies applying /spl lscr//sub p/ measures to basis selection tasks.

Finding the sparsest, or minimum ℓ0-norm, representation of a signal given an overcomplete dictionary of basis vectors is an important prob- lem in many application domains. Unfortunately, the required optimiza- tion problem is often intractable because there is a combinatorial increase in the number of local minima as the number of candidate basis vectors increases. This deficiency has prompted most researchers to instead min- imize surrogate measures, such as the ℓ1-norm, that lead to more tractable computational methods. The downside of this procedure is that we have now introduced a mismatch between our ultimate goal and our objective function. In this paper, we demonstrate a sparse Bayesian learning-based method of minimizing the ℓ0-norm while reducing the number of trou- blesome local minima. Moreover, we derive necessary conditions for local minima to occur via this approach and empirically demonstrate that there are typically many fewer for general problems of interest.

Sparse Bayesian learning and specifically relevance vector machines have received much attention as a means of achieving parsimonious representations of signals in the context of regression and classification. We provide a simplified derivation of this paradigm from a Bayesian evidence perspective and apply it to the problem of basis selection from overcomplete dictionaries. Furthermore, we prove that the stable fixed points of the resulting algorithm are necessarily sparse, providing a solid theoretical justification for adapting the methodology to basis selection tasks. We then include simulation studies comparing sparse Bayesian learning with basis pursuit and the more recent FOCUSS class of basis selection algorithms, empirically demonstrating superior performance in terms of average sparsity and success rate of recovering generative bases.

Recently, relevance vector machines (RVM) have been fashioned from a sparse Bayesian learning (SBL) framework to perform supervised learn- ing using a weight prior that encourages sparsity of representation. The methodology incorporates an additional set of hyperparameters govern- ing the prior, one for each weight, and then adopts a specific approxi- mation to the full marginalization over all weights and hyperparameters. Despite its empirical success however, no rigorous motivation for this particular approximation is currently available. To address this issue, we demonstrate that SBL can be recast as the application of a rigorous vari- ational approximation to the full model by expressing the prior in a dual form. This formulation obviates the necessity of assuming any hyperpri- ors and leads to natural, intuitive explanations of why sparsity is achieved in practice.