One-bit measurements widely exist in the real world, and they can be used to recover sparse signals. This task is known as the problem of learning halfspaces in learning theory and one-bit compressive sensing (1bit-CS) in signal processing. In this paper, we propose novel algorithms based on both convex and nonconvex sparsity-inducing penalties for robust 1bit-CS. We provide a sufficient condition to verify whether a solution is globally optimal or not. Then we show that the globally optimal solution for positive homogeneous penalties can be obtained in two steps: a proximal operator and a normalization step. For several nonconvex penalties, including minimax concave penalty (MCP), $\ell_0$ norm, and sorted $\ell_1$ penalty, we provide fast algorithms for finding the analytical solutions by solving the dual problem. Specifically, our algorithm is more than $200$ times faster than the existing algorithm for MCP. Its efficiency is comparable to the algorithm for the $\ell_1$ penalty in time, while its performance is much better. Among these penalties, the sorted $\ell_1$ penalty is most robust to noise in different settings.
Matrix completion has attracted much interest in the past decade in machine learning and computer vision. For low-rank promotion in matrix completion, the nuclear norm penalty is convenient due to its convexity but has a bias problem. Recently, various algorithms using nonconvex penalties have been proposed, among which the proximal gradient descent (PGD) algorithm is one of the most efficient and effective. For the nonconvex PGD algorithm, whether it converges to a local minimizer and its convergence rate are still unclear. This work provides a nontrivial analysis on the PGD algorithm in the …
The Institute of Statistical Mathematics（统计数理研究所）；The Graduate University for Advanced Science (SOKENDAI)（综合研究大学院大学）；Kyoto University（京都大学）
We consider compressed sensing formulated as a minimization problem of nonconvex sparse penalties, Smoothly Clipped Absolute deviation (SCAD) and Minimax Concave Penalty (MCP). The nonconvexity of these penalties is controlled by nonconvexity parameters, and L1 penalty is contained as a limit with respect to these parameters. The analytically derived reconstruction limit overcomes that of L1 and the algorithmic limit in the Bayes-optimal setting, when the nonconvexity parameters have suitable values. However, for small nonconvexity parameters, where the reconstruction of the relatively dense s…
The University of Chicago（芝加哥大学）；Feinstein Institute of Medical Research（范斯坦医学研究所）；Northwell Health System（诺斯韦尔医疗系统）；Hofstra University（霍夫斯特拉大学）
Regularization methods are often employed in deep learning neural networks (DNNs) to prevent overfitting. For penalty based DNN regularization methods, convex penalties are typically considered because of their optimization guarantees. Recent theoretical work have shown that nonconvex penalties that satisfy certain regularity conditions are also guaranteed to perform well with standard optimization algorithms. In this paper, we examine new and currently existing nonconvex penalties for DNN regularization. We provide theoretical justifications for the new penalties and also assess the performan…
Inria, ENS de Lyon, CNRS, Université Claude Bernard Lyon 1, LIP（法国国家信息与自动化技术研究所、里昂大学、国家科学研究中心、克莱尔伯恩大学、LIP）；IRIT, Université de Toulouse, CNRS（图卢兹大学、国家科学研究中心、IRIT）
Institute of Image Processing and Pattern Recognition, Shanghai Jiao Tong University（上海交通大学图像处理与模式识别研究所）；University of Electronic Science and Technology of China（电子科技大学）；Michigan State University（密歇根州立大学）
Sign information is the key to overcoming the inevitable saturation error in compressive sensing systems, which causes information loss and results in bias. For sparse signal recovery from saturation, we propose to use a linear loss to improve the effectiveness from existing methods that utilize hard constraints/hinge loss for sign consistency. Due to the use of linear loss, an analytical solution in the update progress is obtained, and some nonconvex penalties are applicable, e.g., the minimax concave penalty, the $\ell_0$ norm, and the sorted $\ell_1$ norm. Theoretical analysis reveals that …
Department of Statistical Science, The Graduate University for Advanced Science (SOKENDAI), Hayama-cho, Kanagawa 240-0193, Japan（日本神奈川县叶山町综合研究大学院大学（SOKENDAI）统计科学系）
We investigate the signal reconstruction performance of sparse linear regression in the presence of noise when piecewise continuous nonconvex penalties are used. Among such penalties, we focus on the SCAD penalty. The contributions of this study are three-fold: We first present a theoretical analysis of a typical reconstruction performance, using the replica method, under the assumption that each component of the design matrix is given as an independent and identically distributed (i.i.d.) Gaussian variable. This clarifies the superiority of the SCAD estimator compared with $\ell_1$ in a wide …
Support vector machines (SVMs) with sparsity-inducing nonconvex penalties have received considerable attentions for the characteristics of automatic classification and variable selection. However, it is quite challenging to solve the nonconvex penalized SVMs due to their nondifferentiability, nonsmoothness and nonconvexity. In this paper, we propose an efficient ADMM-based algorithm to the nonconvex penalized SVMs. The proposed algorithm covers a large class of commonly used nonconvex regularization terms including the smooth clipped absolute deviation (SCAD) penalty, minimax concave penalty (…
Nonconvex Zeroth-Order Stochastic ADMM Methods with Lower Function Query Complexity
University of Pittsburgh（匹兹堡大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Duke University（杜克大学）；University of Maryland College Park（马里兰大学帕克分校）
In the past decade, sparse and low-rank recovery have drawn much attention in many areas such as signal/image processing, statistics, bioinformatics and machine learning. To achieve sparsity and/or low-rankness inducing, the $\ell_1$ norm and nuclear norm are of the most popular regularization penalties due to their convexity. While the $\ell_1$ and nuclear norm are convenient as the related convex optimization problems are usually tractable, it has been shown in many applications that a nonconvex penalty can yield significantly better performance. In recent, nonconvex regularization based spa…
College of Electronic and Optical Engineering（电子与光学工程学院）；College of Microelectronics（微电子学院）；Nanjing University of Posts and Telecommunications（南京邮电大学）；College of Telecommunications & Information Engineering（通信与信息工程学院）
Group sparse representation (GSR) based method has led to great successes in various image recovery tasks, which can be converted into a low-rank matrix minimization problem. As a widely used surrogate function of low-rank, the nuclear norm based convex surrogate usually leads to over-shrinking problem, since the standard soft-thresholding operator shrinks all singular values equally. To improve traditional sparse representation based image compressive sensing (CS) performance, we propose a generalized CS framework based on GSR model, which leads to a nonconvex nonsmooth low-rank minimization …
