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

In recent years, low-rank tensor completion (LRTC) problems have received a significant amount of attention in computer vision, data mining, and signal processing. The existing trace norm minimization algorithms for iteratively solving LRTC problems involve multiple singular value decompositions of very large matrices at each iteration. Therefore, they suffer from high computational cost. In this paper, we propose a novel trace norm regularized CANDECOMP/PARAFAC decomposition (TNCP) method for simultaneous tensor decomposition and completion. We first formulate a factor matrix rank minimization model by deducing the relation between the rank of each factor matrix and the mode-n rank of a tensor. Then, we introduce a tractable relaxation of our rank function, and then achieve a convex combination problem of much smaller-scale matrix trace norm minimization. Finally, we develop an efficient algorithm based on alternating direction method of multipliers to solve our problem. The promising experimental results on synthetic and real-world data validate the effectiveness of our TNCP method. Moreover, TNCP is significantly faster than the state-of-the-art methods and scales to larger problems.

Low-rank tensor completion (LRTC) has successfully been applied to a wide range of real-world problems. Despite the broad, successful applications, existing LRTC methods may become very slow or even not applicable for large-scale problems. To address this issue, a novel core tensor trace-norm minimization (CTNM) method is proposed for simultaneous tensor learning and decomposition, and has a much lower computational complexity. In our solution, first, the equivalence relation of trace norm of a low-rank tensor and its core tensor is induced. Second, the trace norm of the core tensor is used to replace that of the whole tensor, which leads to two much smaller scale matrix TNM problems. Finally, an efficient alternating direction augmented Lagrangian method is developed to solve our problems. Our CTNM formulation needs only O((R ^N + NRI) log(√I ^N )) observations to reliably recover an Nth-order I × I ×⋯× I tensor of n-rank (r, r, .. ., r), compared with O(rI ^N-1 ) observations required by those tensor TNM methods (I ≫ R ≥ r). Extensive experimental results show that CTNM is usually more accurate than them, and is orders of magnitude faster.

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

Pixel-wise classification plays an important role for image interpretation. The remote sensing images, especially polarimetric synthetic aperture radar (PolSAR), have provided wide applications for both military and civilian users regardless of weather or lighting conditions. However, the classification of heterogeneous imagery is still challenging. In this paper, we propose a novel deep fuzzy graph convolutional network (DFGCN) for pixel-wise classification of PolSAR imagery. Inspired by the fuzzy logic, the imagery (i.e., a reflection of the backscattering of the ground) is represented by a fuzzy graph, whose node is associated with a feature vector, and fuzzy weights denoting the similarity degrees between nodes are defined by using both feature distance and spatial relation. We also construct a new graph residual module stacking multiple residual layers to extend the depth of our fuzzy graph network. Moreover, we present a graph shrinking strategy to reduce the graph size for training and predicting the labels of large-scale PolSAR imagery, which can cut down the computational cost. The experimental results on various real-world PolSAR datasets indicate that DFGCN dramatically enhances classification accuracies on both heterogeneous and homogeneous regions with limited labeled pixels compared with the state-of-the-art methods.

Deep neural networks are vulnerable to adversarial examples crafted by applying human-imperceptible perturbations on clean inputs. Although many attack methods can achieve high success rates in the white-box setting, they also exhibit weak transferability in the black-box setting. Recently, various methods have been proposed to improve adversarial transferability, in which the input transformation is one of the most effective methods. In this work, we notice that existing input transformation-based works mainly adopt the transformed data in the same domain for augmentation. Inspired by domain generalization, we aim to further improve the transferability using the data augmented from different domains. Specifically, a style transfer network can alter the distribution of low-level visual features in an image while preserving semantic content for humans. Hence, we propose a novel attack method named Style Transfer Method (STM) that utilizes a proposed arbitrary style transfer network to transform the images into different domains. To avoid inconsistent semantic information of stylized images for the classification network, we fine-tune the style transfer network and mix up the generated images added by random noise with the original images to maintain semantic consistency and boost input diversity. Extensive experimental results on the ImageNet-compatible dataset show that our proposed method can significantly improve the adversarial transferability on either normally trained models or adversarially trained models than state-of-the-art input transformation-based attacks. Code is available at: https://github.com/Zhijin-Ge/STM.