Schatten-$p$ Quasi-Norm Regularized Matrix Optimization via Iterative Reweighted Singular Value Minimization

Abstract

In this paper we study general Schatten-$p$ quasi-norm (SPQN) regularized matrix minimization problems. In particular, we first introduce a class of first-order stationary points for them, and show that the first-order stationary points introduced in [11] for an SPQN regularized $vector$ minimization problem are equivalent to those of an SPQN regularized $matrix$ minimization reformulation. We also show that any local minimizer of the SPQN regularized matrix minimization problems must be a first-order stationary point. Moreover, we derive lower bounds for nonzero singular values of the first-order stationary points and hence also of the local minimizers of the SPQN regularized matrix minimization problems. The iterative reweighted singular value minimization (IRSVM) methods are then proposed to solve these problems, whose subproblems are shown to have a closed-form solution. In contrast to the analogous methods for the SPQN regularized $vector$ minimization problems, the convergence analysis of these methods is significantly more challenging. We develop a novel approach to establishing the convergence of these methods, which makes use of the expression of a specific solution of their subproblems and avoids the intricate issue of finding the explicit expression for the Clarke subdifferential of the objective of their subproblems. In particular, we show that any accumulation point of the sequence generated by the IRSVM methods is a first-order stationary point of the problems. Our computational results demonstrate that the IRSVM methods generally outperform some recently developed state-of-the-art methods in terms of solution quality and/or speed.

This paper has been withdrawn by the author due to major revision and corrections

References in corpus (2)

• Nearly unbiased variable selection under minimax concave penalty
• Iterative Reweighted Minimization Methods for $l_p$ Regularized Unconstrained Nonlinear Programming

Cited by in corpus (5)

• Unified Scalable Equivalent Formulations for Schatten Quasi-Norms
• Tractable and Scalable Schatten Quasi-Norm Approximations for Rank Minimization
• Matrix Completion via Nonconvex Regularization: Convergence of the Proximal Gradient Algorithm
• A Survey on Nonconvex Regularization Based Sparse and Low-Rank Recovery in Signal Processing, Statistics, and Machine Learning
• Proximal linearized iteratively reweighted least squares for a class of nonconvex and nonsmooth problems