Hierarchical classification aims to sort the object into a hierarchical structure of categories. For example, a bird can be categorized according to a three-level hierarchy of order, family, and species. Existing methods commonly address hierarchical classification by decoupling it into a series of multi-class classification tasks. However, such a multi-task learning strategy fails to fully exploit the correlation among various categories across different levels of the hierarchy. In this paper, we propose Label Hierarchy Transition (LHT), a unified probabilistic framework based on deep learning, to address the challenges of hierarchical classification. The LHT framework consists of a transition network and a confusion loss. The transition network focuses on explicitly learning the label hierarchy transition matrices, which has the potential to effectively encode the underlying correlations embedded within class hierarchies. The confusion loss encourages the classification network to learn correlations across different label hierarchies during training. The proposed framework can be readily adapted to any existing deep network with only minor modifications. We experiment with a series of public benchmark datasets for hierarchical classification problems, and the results demonstrate the superiority of our approach beyond current state-of-the-art methods. Furthermore, we extend our proposed LHT framework to the skin lesion diagnosis task and validate its great potential in computer-aided diagnosis.

In online continual learning (CL), models trained on changing distributions easily forget previously learned knowledge and bias toward newly received tasks. To address this issue, we present Continual Bias Adaptor (CBA), a bi-level framework that augments the classification network to adapt to catastrophic distribution shifts during training, achieving a stable consolidation of all seen tasks. However, CBA adjusts distribution shifts in a class-specific manner, exacerbating the stability gap issue and fails to meet the need for continual testing to some extent. To mitigate this challenge, we further propose a novel class-agnostic CBA module that separately aggregates the posterior probabilities of new and old tasks, applying a stable adjustment to the results. We combine these two kinds of CBA modules into a unified Dual-CBA module, which thus is capable of adapting to catastrophic distribution shifts and simultaneously meets the real-time testing requirements of online CL. Besides, we propose Incremental Batch Normalization (IBN), a tailored BN module to re-estimate its population statistics for alleviating the feature bias arising from our bi-level framework. We theoretically provide some insights into how it mitigates distribution shifts, and empirically demonstrate its superiority through extensive experiments based on four rehearsal-based baselines and three public CL benchmarks.

Pseudo-labeling is a commonly used paradigm in semi-supervised learning, yet its application to semi-supervised regression (SSR) remains relatively under-explored. Unlike classification, where pseudo-labels are discrete and confidence-based filtering is effective, SSR involves continuous outputs with heteroscedastic noise, making it challenging to assess pseudo-label reliability. As a result, naive pseudo-labeling can lead to error accumulation and overfitting to incorrect labels. To address this, we propose an uncertainty-aware pseudo-labeling framework that dynamically adjusts pseudo-label influence from a bi-level optimization perspective. By jointly minimizing empirical risk over all data and optimizing uncertainty estimates to enhance generalization on labeled data, our method effectively mitigates the impact of unreliable pseudo-labels. We provide theoretical insights and extensive experiments to validate our approach across various benchmark SSR datasets, and the results demonstrate superior robustness and performance compared to existing methods.

This paper focuses on the optimization of overparameterized, non-convex low-rank matrix sensing (LRMS)—an essential component in contemporary statistics and machine learning. Recent years have witnessed significant breakthroughs in first-order methods, such as gradient descent, for tackling this non-convex optimization problem. However, the presence of numerous saddle points often prolongs the time required for gradient descent to overcome these obstacles. Moreover, overparameterization can markedly decelerate gradient descent methods, transitioning its convergence rate from linear to sub-linear. In this paper, we introduce an approximated Gauss-Newton (AGN) method for tackling the non-convex LRMS problem. Notably, AGN incurs a computational cost comparable to gradient descent per iteration but converges much faster without being slowed down by saddle points. We prove that, despite the non-convexity of the objective function, AGN achieves Q-linear convergence from random initialization to the global optimal solution. The global Q-linear convergence of AGN represents a substantial enhancement over the convergence of the existing methods for the overparameterized non-convex LRMS. The code for this paper is available at \url{https://github.com/hsijiaxidian/AGN}.

Abstract. Recently, the CTV-RPCA model proposed the first recoverable theory for separating low-rank and local-smooth matrices and sparse matrices based on the correlated total variation (CTV) regularizer. However, the CTV-RPCA model ignores the influence of noise, which makes the model unable to effectively extract low-rank and local-smooth principal components under noisy circumstances. To alleviate this issue, this article extends the CTV-RPCA model by considering the influence of noise and proposes two robust models with parameter adaptive adjustment, i.e., Stable Principal Component Pursuit based on CTV (CTV-SPCP) and Square Root Principal Component Pursuit based on CTV (CTV-[Formula: see text]). Furthermore, we present a statistical recoverable error bound for the proposed models, which allows us to know the relationship between the solution of the proposed models and the ground-truth. It is worth mentioning that, in the absence of noise, our theory degenerates back to the exact recoverable theory of the CTV-RPCA model. Finally, we develop the effective algorithms with the strict convergence guarantees. Extensive experiments adequately validate the theoretical assertions and also demonstrate the superiority of the proposed models over many state-of-the-art methods on various typical applications, including video foreground extraction, multispectral image denoising, and hyperspectral image denoising. The source code is released at https://github.com/andrew-pengjj/CTV-SPCP .

Weight decay (WD) is a fundamental and practical regularization technique in improving generalization of current deep learning models. However, it is observed that the WD does not work effectively for an adaptive optimization algorithm (such as Adam), as it works for SGD. Specifically, the solution found by Adam with the WD often generalizes unsatisfactorily. Though efforts have been made to mitigate this issue, the reason for such deficiency is still vague. In this article, we first show that when using the Adam optimizer, the weight norm increases very fast along with the training procedure, which is in contrast to SGD where the weight norm increases relatively slower and tends to converge. The fast increase of weight norm is adverse to WD; in consequence, the Adam optimizer will lose efficacy in finding solution that generalizes well. To resolve this problem, we propose to tailor Adam by introducing a regularization term on the adaptive learning rate, such that it is friendly to WD. Meanwhile, we introduce first moment on the WD to further enhance the regularization effect. We show that the proposed method is able to find solution with small norm and generalizes better than SGD. We test the proposed method on general image classification and fine-grained image classification tasks with different networks. Experimental results on all these cases substantiate the effectiveness of the proposed method in help improving the generalization. Specifically, the proposed method improves the test accuracy of Adam by a large margin and even improves the performance of SGD by 0.84% on CIFAR 10 and 1.03 % on CIFAR 100 with ResNet-50. The code of this article is public available at xxx.

Online continual learning (CL) aims to learn new knowledge and consolidate previously learned knowledge from non-stationary data streams. Due to the time-varying training setting, the model learned from a changing distribution easily forgets the previously learned knowledge and biases toward the newly received task. To address this problem, we propose a Continual Bias Adaptor (CBA) module to augment the classifier network to adapt to catastrophic distribution change during training, such that the classifier network is able to learn a stable consolidation of previously learned tasks. In the testing stage, CBA can be removed which introduces no additional computation cost and memory overhead. We theoretically reveal the reason why the proposed method can effectively alleviate catastrophic distribution shifts, and empirically demonstrate its effectiveness through extensive experiments based on four rehearsal-based baselines and three public continual learning benchmarks1.

Low-rank matrix factorization (LRMF) is a canonical problem in non-convex optimization, the objective function to be minimized is non-convex and even non-smooth, which makes the global convergence guarantee of gradient-based algorithm quite challenging. Recent work made a breakthrough on proving that standard gradient descent converges to the $\varepsilon$-global minima after $O( \frac{d \kappa^2}{\tau^2} {\rm ln} \frac{d \sigma_d}{\tau} + \frac{d \kappa^2}{\tau^2} {\rm ln} \frac{\sigma_d}{\varepsilon})$ iterations from small initialization with a very small learning rate (both are related to the small constant $\tau$). While the dependence of the convergence on the \textit{condition number} $\kappa$ and \textit{small learning rate} makes it not practical especially for ill-conditioned LRMF problem. In this paper, we show that precondition helps in accelerating the convergence and prove that the scaled gradient descent (ScaledGD) and its variant, alternating scaled gradient descent (AltScaledGD) converge to an $\varepsilon$-global minima after $O( {\rm ln} \frac{d}{\delta} + {\rm ln} \frac{d}{\varepsilon})$ iterations from general random initialization. Meanwhile, for small initialization as in gradient descent, both ScaledGD and AltScaledGD converge to $\varepsilon$-global minima after only $O({\rm ln} \frac{d}{\varepsilon})$ iterations. Furthermore, we prove that as a proximity to the alternating minimization, AltScaledGD converges faster than ScaledGD, its global convergence does not rely on small learning rate and small initialization, which certificates the advantages of AltScaledGD in LRMF.

The deep unfolding network (DUN) provides an efficient framework for image restoration. It consists of a regularization module and a data fitting module. In existing DUN models, it is common to directly use a deep convolution neural network (DCNN) as the regularization module, and perform data fitting before regularization in each iteration/stage. In this work, we present a DUN by incorporating a new regularization module, and putting the regularization module before the data fitting module. The proposed regularization model is deducted by using the regularization by denoing (RED) and plugging in it a newly designed DCNN. For the data fitting module, we use the closed-form solution with Faster Fourier Transform (FFT). The resulted DRED-DUN model has some major advantages. First, the regularization model inherits the flexibility of learned image-adaptive and interpretability of RED. Second, the DRED-DUN model is an end-to-end trainable DUN, which learns the regularization network and other parameters jointly, thus leads to better restoration performance than the plug-and-play framework. Third, extensive experiments show that, our proposed model significantly outperforms the-state-of-the-art model-based methods and learning based methods in terms of PSNR indexes as well as the visual effects. In particular, our method has much better capability in recovering salient image components such as edges and small scale textures.

Spectral regularization is a widely used approach for low-rank matrix recovery (LRMR) by regularizing matrix singular values. Most of the existing LRMR solvers iteratively compute the singular values via applying singular value decomposition (SVD) on a dense matrix, which is computationally expensive and severely limits their applications to large-scale problems. To address this issue, we present a generalized unitarily invariant gauge (GUIG) function for LRMR. The proposed GUIG function does not act on the singular values; however, we show that it generalizes the well-known spectral functions, including the rank function, the Schatten- p quasi-norm, and logsum of singular values. The proposed GUIG regularization model can be formulated as a bilinear variational problem, which can be efficiently solved without computing SVD. Such a property makes it well suited for large-scale LRMR problems. We apply the proposed GUIG model to matrix completion and robust principal component analysis and prove the convergence of the algorithms. Experimental results demonstrate that the proposed GUIG method is not only more accurate but also much faster than the state-of-the-art algorithms, especially on large-scale problems.

Deep convolutional neural networks (DCNN) have been successfully used in many low-level vision problems such as image denoising. Recent studies on the mathematical foundation of DCNN has revealed that the forward propagation of DCNN corresponds to a dynamic system, which can be described by an ordinary differential equation (ODE) and solved by the optimal control method. However, most of these methods employ integer-order differential equation, which has local connectivity in time space and cannot describe the long-term memory of the system. Inspired by the fact that the fractional-order differential equation has long-term memory, in this paper we develop an advanced image denoising network, namely FOCNet, by solving a fractional optimal control (FOC) problem. Specifically, the network structure is designed based on the discretization of a fractional-order differential equation, which enjoys long-term memory in both forward and backward passes. Besides, multi-scale feature interactions are introduced into the FOCNet to strengthen the control of the dynamic system. Extensive experiments demonstrate the leading performance of the proposed FOCNet on image denoising. Code will be made available.

Edge-preserving image smoothing is an important step for many low-level vision problems. Though many algorithms have been proposed, there are several difficulties hindering its further development. First, most existing algorithms cannot perform well on a wide range of image contents using a single parameter setting. Second, the performance evaluation of edge-preserving image smoothing remains subjective, and there lacks a widely accepted datasets to objectively compare the different algorithms. To address these issues and further advance the state of the art, in this work we propose a benchmark for edge-preserving image smoothing. This benchmark includes an image dataset with groundtruth image smoothing results as well as baseline algorithms that can generate competitive edge-preserving smoothing results for a wide range of image contents. The established dataset contains 500 training and testing images with a number of representative visual object categories, while the baseline methods in our benchmark are built upon representative deep convolutional network architectures, on top of which we design novel loss functions well suited for edge-preserving image smoothing. The trained deep networks run faster than most state-of-the-art smoothing algorithms with leading smoothing results both qualitatively and quantitatively. The benchmark will be made publicly accessible.

In this paper, we propose an adaptive regularizer learning method in the framework of MAP for low rank approximation (ARLLR). We assume that the prior distribution of the singular values is Laplacian with varying scale parameters. By using a full maximize a posterior (MAP) we learn the optimal scale parameters iteratively. We indicate that ARLLR is equivalent to low rank approximation regularized by Logarithm on singular values. In theory, we prove that ARLLR (Logarithm regularization) although being non-convex can be solved in closed form, and we further prove that local minimum can be easily obtained. Finally, ARLLR is applied to image de-noising. Experimental results show that the proposed method enhances image denoising compared with state-of-the-art image denoising algorithms(especially for BM3D, SAIST and WNNM) in both quantity value (PSNR) and visual quality.