An exact penalty method for semidefinite-box-constrained low-rank matrix optimization problems

2018 ◽  
Vol 40 (1) ◽  
pp. 563-586
Author(s):  
Tianxiang Liu ◽  
Zhaosong Lu ◽  
Xiaojun Chen ◽  
Yu-Hong Dai
Keyword(s):  
Penalty Method ◽  
Original Problem ◽  
Optimization Problems ◽  
Rank Correlation ◽  
Low Rank ◽  
Matrix Optimization ◽  
Rank Constraint ◽  
Rank Matrix ◽  
Adaptive Penalty ◽  
Low Rank Matrix

Abstract This paper considers a matrix optimization problem where the objective function is continuously differentiable and the constraints involve a semidefinite-box constraint and a rank constraint. We first replace the rank constraint by adding a non-Lipschitz penalty function in the objective and prove that this penalty problem is exact with respect to the original problem. Next, for the penalty problem we present a nonmonotone proximal gradient (NPG) algorithm whose subproblem can be solved by Newton’s method with globally quadratic convergence. We also prove the convergence of the NPG algorithm to a first-order stationary point of the penalty problem. Furthermore, based on the NPG algorithm, we propose an adaptive penalty method (APM) for solving the original problem. Finally, the efficiency of an APM is shown via numerical experiments for the sensor network localization problem and the nearest low-rank correlation matrix problem.

scholarly journals On critical points of quadratic low-rank matrix optimization problems

2020 ◽  
Vol 40 (4) ◽  
pp. 2626-2651
Author(s):  
André Uschmajew ◽  
Bart Vandereycken
Keyword(s):  
Critical Points ◽  
Optimization Problems ◽  
Quadratic Function ◽  
Singular Values ◽  
Positive Semidefinite ◽  
Optimization Methods ◽  
Low Rank ◽  
Local Minima ◽  
Rank Matrix ◽  
Low Rank Matrix

Abstract The absence of spurious local minima in certain nonconvex low-rank matrix recovery problems has been of recent interest in computer science, machine learning and compressed sensing since it explains the convergence of some low-rank optimization methods to global optima. One such example is low-rank matrix sensing under restricted isometry properties (RIPs). It can be formulated as a minimization problem for a quadratic function on the Riemannian manifold of low-rank matrices, with a positive semidefinite Riemannian Hessian that acts almost like an identity on low-rank matrices. In this work new estimates for singular values of local minima for such problems are given, which lead to improved bounds on RIP constants to ensure absence of nonoptimal local minima and sufficiently negative curvature at all other critical points. A geometric viewpoint is taken, which is inspired by the fact that the Euclidean distance function to a rank-$k$ matrix possesses no critical points on the corresponding embedded submanifold of rank-$k$ matrices except for the single global minimum.

scholarly journals Radar Target and Moving Clutter Separation Based on the Low-Rank Matrix Optimization

2018 ◽  
Vol 56 (8) ◽  
pp. 4765-4780 ◽  
Author(s):  
Jiapeng Yin ◽  
Christine Unal ◽  
Marc Schleiss ◽  
Herman Russchenberg
Keyword(s):  
Low Rank ◽  
Radar Target ◽  
Matrix Optimization ◽  
Rank Matrix ◽  
Low Rank Matrix

scholarly journals s-Goodness for Low-Rank Matrix Recovery

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Lingchen Kong ◽  
Levent Tunçel ◽  
Naihua Xiu
Keyword(s):  
Linear Transformation ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Nuclear Norm ◽  
Low Rank ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
Necessary And Sufficient ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

Low-rank matrix recovery (LMR) is a rank minimization problem subject to linear equality constraints, and it arises in many fields such as signal and image processing, statistics, computer vision, and system identification and control. This class of optimization problems is generally𝒩𝒫hard. A popular approach replaces the rank function with the nuclear norm of the matrix variable. In this paper, we extend and characterize the concept ofs-goodness for a sensing matrix in sparse signal recovery (proposed by Juditsky and Nemirovski (Math Program, 2011)) to linear transformations in LMR. Using the two characteristics-goodness constants,γsandγ^s, of a linear transformation, we derive necessary and sufficient conditions for a linear transformation to bes-good. Moreover, we establish the equivalence ofs-goodness and the null space properties. Therefore,s-goodness is a necessary and sufficient condition for exacts-rank matrix recovery via the nuclear norm minimization.

scholarly journals Low-rank Matrix Optimization for Video Segmentation Research

2017 ◽  
Vol 6 (1) ◽  
pp. 36
Author(s):  
Caiyun Huang ◽  
Guojun Qin
Keyword(s):  
Video Segmentation ◽  
Optimal Solution ◽  
Low Rank ◽  
Superior Performance ◽  
Closed Form Solutions ◽  
Matrix Optimization ◽  
Rank Matrix ◽  
Low Rank Representation ◽  
Spatio Temporal ◽  
Low Rank Matrix

This paper investigates how to perform robust and efficient unsupervised video segmentation while suppressing the effects of data noises and/or corruptions. The low-rank representation is pursued for video segmentation. The supervoxels affinity matrix of an observed video sequence is given, low-rank matrix optimization seeks a optimal solution by making the matrix rank explicitly determined. We iteratively optimize them with closed-form solutions. Moreover, we incorporate a discriminative replication prior into our framework based on the obervation that small-size video patterns, and it tends to recur frequently within the same object. The video can be segmented into several spatio-temporal regions by applying the Normalized-Cut algorithm with the solved low-rank representation. To process the streaming videos, we apply our algorithm sequentially over a batch of frames over time, in which we also develop several temporal consistent constraints improving the robustness. Extensive experiments are on the public benchmarks, they demonstrate superior performance of our framework over other approaches．

Active Subspace: Toward Scalable Low-Rank Learning

2012 ◽  
Vol 24 (12) ◽  
pp. 3371-3394 ◽  
Author(s):  
Guangcan Liu ◽  
Shuicheng Yan
Keyword(s):  
Large Scale ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Low Rank ◽  
Small Scale ◽  
Large Solution ◽  
Rank Matrix ◽  
Low Rank Matrix ◽  
Active Subspace ◽  
Solution Matrix

We address the scalability issues in low-rank matrix learning problems. Usually these problems resort to solving nuclear norm regularized optimization problems (NNROPs), which often suffer from high computational complexities if based on existing solvers, especially in large-scale settings. Based on the fact that the optimal solution matrix to an NNROP is often low rank, we revisit the classic mechanism of low-rank matrix factorization, based on which we present an active subspace algorithm for efficiently solving NNROPs by transforming large-scale NNROPs into small-scale problems. The transformation is achieved by factorizing the large solution matrix into the product of a small orthonormal matrix (active subspace) and another small matrix. Although such a transformation generally leads to nonconvex problems, we show that a suboptimal solution can be found by the augmented Lagrange alternating direction method. For the robust PCA (RPCA) (Candès, Li, Ma, & Wright, 2009 ) problem, a typical example of NNROPs, theoretical results verify the suboptimality of the solution produced by our algorithm. For the general NNROPs, we empirically show that our algorithm significantly reduces the computational complexity without loss of optimality.

scholarly journals Global Optimality in Low-Rank Matrix Optimization

2018 ◽  
Vol 66 (13) ◽  
pp. 3614-3628 ◽  
Author(s):  
Zhihui Zhu ◽  
Qiuwei Li ◽  
Gongguo Tang ◽  
Michael B. Wakin
Keyword(s):  
Low Rank ◽  
Global Optimality ◽  
Matrix Optimization ◽  
Rank Matrix ◽  
Low Rank Matrix

scholarly journals The Global Optimization Geometry of Low-Rank Matrix Optimization

2021 ◽  
pp. 1-1
Author(s):  
Zhihui Zhu ◽  
Qiuwei Li ◽  
Gongguo Tang ◽  
Michael B. Wakin
Keyword(s):  
Global Optimization ◽  
Low Rank ◽  
Matrix Optimization ◽  
Rank Matrix ◽  
Low Rank Matrix
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