In recent years, the nuclear norm minimization (NNM) problem has been attracting much attention in computer vision and machine learning. The NNM problem is capitalized on its convexity and it can be solved efficiently. The standard nuclear norm regularizes all singular values equally, which is however not flexible enough to fit real scenarios. Weighted nuclear norm minimization (WNNM) is a natural extension and generalization of NNM. By assigning properly different weights to different singular values, WNNM can lead to state-of-the-art results in applications such as image denoising. Nevertheless, so far the global optimal solution of WNNM problem is not completely solved yet due to its non-convexity in general cases. In this article, we study the theoretical properties of WNNM and prove that WNNM can be equivalently transformed into a quadratic programming problem with linear constraints. This implies that WNNM is equivalent to a convex problem and its global optimum can be readily achieved by off-the-shelf convex optimization solvers. We further show that when the weights are non-descending, the globally optimal solution of WNNM can be obtained in closed-form.
Many applications require recovering a ground truth low-rank matrix from noisy observations of the entries, which in practice is typically formulated as a weighted low-rank approximation problem and solved by non-convex optimization heuristics such as alternating minimization. In this paper, we provide provable recovery guarantee of weighted low-rank via a simple alternating minimization algorithm. In particular, for a natural class of matrices and weights and without any assumption on the noise, we bound the spectral norm of the difference between the recovered matrix and the ground truth, by…
Nonlocal image representation or group sparsity has attracted considerable interest in various low-level vision tasks and has led to several state-of-the-art image denoising techniques, such as BM3D, LSSC. In the past, convex optimization with sparsity-promoting convex regularization was usually regarded as a standard scheme for estimating sparse signals in noise. However, using convex regularization can not still obtain the correct sparsity solution under some practical problems including image inverse problems. In this paper we propose a non-convex weighted $\ell_p$ minimization based group …
University of Torontoland（多伦多兰大学）；Googol ShallowMind（谷歌浅智）
Low rank matrix approximation (LRMA) has drawn increasing attention in recent years, due to its wide range of applications in computer vision and machine learning. However, LRMA, achieved by nuclear norm minimization (NNM), tends to over-shrink the rank components with the same threshold and ignore the differences between rank components. To address this problem, we propose a flexible and precise model named multi-band weighted $l_p$ norm minimization (MBWPNM). The proposed MBWPNM not only gives more accurate approximation with a Schatten $p$-norm, but also considers the prior knowledge where …
The Hong Kong Polytechnic University（香港理工大学）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；Harbin Institute of Technology（哈尔滨工业大学）
Low rank matrix approximation (LRMA), which aims to recover the underlying low rank matrix from its degraded observation, has a wide range of applications in computer vision. The latest LRMA methods resort to using the nuclear norm minimization (NNM) as a convex relaxation of the nonconvex rank minimization. However, NNM tends to over-shrink the rank components and treats the different rank components equally, limiting its flexibility in practical applications. We propose a more flexible model, namely the Weighted Schatten $p$-Norm Minimization (WSNM), to generalize the NNM to the Schatten $p$…
University of Electronic Science and Technology of China（中国电子科技大学）
Recovering a large matrix from limited measurements is a challenging task arising in many real applications, such as image inpainting, compressive sensing and medical imaging, and this kind of problems are mostly formulated as low-rank matrix approximation problems. Due to the rank operator being non-convex and discontinuous, most of the recent theoretical studies use the nuclear norm as a convex relaxation and the low-rank matrix recovery problem is solved through minimization of the nuclear norm regularized problem. However, a major limitation of nuclear norm minimization is that all the sin…
The Hong Kong Polytechnic University（香港理工大学）；Xidian University（西安电子科技大学）
Most of the existing denoising algorithms are developed for grayscale images, while it is not a trivial work to extend them for color image denoising because the noise statistics in R, G, B channels can be very different for real noisy images. In this paper, we propose a multi-channel (MC) optimization model for real color image denoising under the weighted nuclear norm minimization (WNNM) framework. We concatenate the RGB patches to make use of the channel redundancy, and introduce a weight matrix to balance the data fidelity of the three channels in consideration of their different noise sta…
This paper presents a patch-wise low-rank based image denoising method with constrained variational model involving local and nonlocal regularization. On one hand, recent patch-wise methods can be represented as a low-rank matrix approximation problem whose convex relaxation usually depends on nuclear norm minimization (NNM). Here, we extend the NNM to the nonconvex schatten p-norm minimization with additional weights assigned to different singular values, which is referred to as the Weighted Schatten p-Norm Minimization (WSNM). An efficient algorithm is also proposed to solve the WSNM problem…
In image denoising (IDN) processing, the low-rank property is usually considered as an important image prior. As a convex relaxation approximation of low rank, nuclear norm based algorithms and their variants have attracted significant attention. These algorithms can be collectively called image domain based methods, whose common drawback is the requirement of great number of iterations for some acceptable solution. Meanwhile, the sparsity of images in a certain transform domain has also been exploited in image denoising problems. Sparsity transform learning algorithms can achieve extremely fa…
University of Electronic Science and Technology of China（电子科技大学）；Nanyang Technological University（南洋理工大学）；Nanjing Institute of Technology（南京工程学院）；University of Macau（澳门大学）
The nuclear norm minimization (NNM) is commonly used to approximate the matrix rank by shrinking all singular values equally. However, the singular values have clear physical meanings in many practical problems, and NNM may not be able to faithfully approximate the matrix rank. To alleviate the above-mentioned limitation of NNM, recent studies have suggested that the weighted nuclear norm minimization (WNNM) can achieve a better rank estimation than NNM, which heuristically set the weight being inverse to the singular values. However, it still lacks a rigorous explanation why WNNM is more effe…
Many practical applications such as gene expression analysis, multi-task learning, image recognition, signal processing, and medical data analysis pursue a sparse solution for the feature selection purpose and particularly favor the nonzeros \emph{evenly} distributed in different groups. The exclusive sparsity norm has been widely used to serve to this purpose. However, it still lacks systematical studies for exclusive sparsity norm optimization. This paper offers two main contributions from the optimization perspective: 1) We provide several efficient algorithms to solve exclusive sparsity no…
Since the matrix formed by nonlocal similar patches in a natural image is of low rank, the nuclear norm minimization (NNM) has been widely used in various image processing studies. Nonetheless, nuclear norm based convex surrogate of the rank function usually over-shrinks the rank components and makes different components equally, and thus may produce a result far from the optimum. To alleviate the above-mentioned limitations of the nuclear norm, in this paper we propose a new method for image restoration via the non-convex weighted Lp nuclear norm minimization (NCW-NNM), which is able to more …
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Pontificia Universidad Católica de Chile（智利天主教大学）
Composite minimization is a powerful framework in large-scale convex optimization, based on decoupling of the objective function into terms with structurally different properties and allowing for more flexible algorithmic design. We introduce a new algorithmic framework for complementary composite minimization, where the objective function decouples into a (weakly) smooth and a uniformly convex term. This particular form of decoupling is pervasive in statistics and machine learning, due to its link to regularization. The main contributions of our work are summarized as follows. First, we intro…
Quaternion Nuclear Norm minus Frobenius Norm Minimization for color image reconstruction
School of Mathematical Science, Inner Mongolia University（内蒙古大学数学科学学院）；The Chinese University of Hong Kong（香港中文大学）；Hong Kong Baptist University（香港浸会大学）
Nanjing University（南京大学）；Nokia Bell Labs（诺基亚贝尔实验室）；University of Oulu（奥卢大学）；University of Electronic Science and Technology of China（电子科技大学）
Rank minimization methods have attracted considerable interest in various areas, such as computer vision and machine learning. The most representative work is nuclear norm minimization (NNM), which can recover the matrix rank exactly under some restricted and theoretical guarantee conditions. However, for many real applications, NNM is not able to approximate the matrix rank accurately, since it often tends to over-shrink the rank components. To rectify the weakness of NNM, recent advances have shown that weighted nuclear norm minimization (WNNM) can achieve a better matrix rank approximation …
The Chinese University of Hong Kong（香港中文大学）；Shenzhen Research Institute of Big Data, the Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）深圳大数据研究院）；University of Minnesota（明尼苏达大学）；Peking University（北京大学）；Shanghai Jiao Tong University（上海交通大学）
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 lp-norm minimization with two specific values of p, i.e., p=1/2 and p=2/3, we propose two…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Nanjing Hydraulic Research Institute（南京水利科学研究院）；State Key Laboratory of Urban Water Resource and Environment（城市水资源与水环境国家重点实验室）；The Key Laboratory of Rivers and Lakes Governance and Flood Protection of Yangtse River Water Conservancy Committee（长江水利委员会江湖治理与防洪重点实验室）
Subspace clustering refers to the problem of segmenting a set of data points approximately drawn from a union of multiple linear subspaces. Aiming at the subspace clustering problem, various subspace clustering algorithms have been proposed and low rank representation based subspace clustering is a very promising and efficient subspace clustering algorithm. Low rank representation method seeks the lowest rank representation among all the candidates that can represent the data points as linear combinations of the bases in a given dictionary. Nuclear norm minimization is adopted to minimize the …
It is an efficient and effective strategy to utilize the nuclear norm approximation to learn low-rank matrices, which arise frequently in machine learning and computer vision. So the exploration of nuclear norm minimization problems is gaining much attention recently. In this paper we shall prove that the following Low-Rank Representation (LRR) \cite{icml_2010_lrr,lrr_extention} problem: {eqnarray*} \min_{Z} \norm{Z}_*, & {s.t.,} & X=AZ, {eqnarray*} has a unique and closed-form solution, where $X$ and $A$ are given matrices. The proof is based on proving a lemma that allows us to get closed-fo…
Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）；Technical University of Denmark（丹麦技术大学）；Vrije Universiteit Brussel（布鲁塞尔自由大学）；iMinds
In the context of compressed sensing (CS), this paper considers the problem of reconstructing sparse signals with the aid of other given correlated sources as multiple side information. To address this problem, we theoretically study a generic \textcolor{black}{weighted $n$-$\ell_{1}$ minimization} framework and propose a reconstruction algorithm that leverages multiple side information signals (RAMSI). The proposed RAMSI algorithm computes adaptively optimal weights among the side information signals at every reconstruction iteration. In addition, we establish theoretical bounds on the number…
The robust principal component analysis (RPCA), which aims to estimate underlying low-rank and sparse structures from the degraded observation data, has found wide applications in computer vision. It is usually replaced by the principal component pursuit (PCP) model in order to pursue the convex property, leading to the undesirable overshrink problem. In this paper, we propose a dual weighted lp-norm (DWLP) model with a more reasonable weighting rule and weaker powers, which greatly generalizes the previous work and provides a better approximation to the rank minimization problem for original …
The University of Adelaide（阿德莱德大学）；Nanjing University of Science and Technology（南京理工大学）
Minimization of the $L_\infty$ norm, which can be viewed as approximately solving the non-convex least median estimation problem, is a powerful method for outlier removal and hence robust regression. However, current techniques for solving the problem at the heart of $L_\infty$ norm minimization are slow, and therefore cannot scale to large problems. A new method for the minimization of the $L_\infty$ norm is presented here, which provides a speedup of multiple orders of magnitude for data with high dimension. This method, termed Fast $L_\infty$ Minimization, allows robust regression to be app…
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Peng Cheng Laboratory（鹏城实验室）；Huawei Technologies（华为技术有限公司）
Due to the domain discrepancy in visual domain adaptation, the performance of source model degrades when bumping into the high data density near decision boundary in target domain. A common solution is to minimize the Shannon Entropy to push the decision boundary away from the high density area. However, entropy minimization also leads to severe reduction of prediction diversity, and unfortunately brings harm to the domain adaptation. In this paper, we investigate the prediction discriminability and diversity by studying the structure of the classification output matrix of a randomly selected …
