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Schattenp QuasiNorm Regularized Matrix Optimization via Iterative Reweighted Singular Value Minimization
, 2015
"... In this paper we study general Schattenp quasinorm (SPQN) regularized matrix minimization problems. In particular, we first introduce a class of firstorder stationary points for them, and show that the firstorder stationary points introduced in [11] for an SPQN regularized vector minimization ..."
Abstract

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In this paper we study general Schattenp quasinorm (SPQN) regularized matrix minimization problems. In particular, we first introduce a class of firstorder stationary points for them, and show that the firstorder 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 firstorder stationary point. Moreover, we derive lower bounds for nonzero singular values of the firstorder 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 closedform 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