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**1 - 2**of**2**### Schatten-p Quasi-Norm Regularized Matrix Optimization via Iterative Reweighted Singular Value Minimization∗

, 2015

"... In this paper we study general Schatten-p quasi-norm (SPQN) regularized matrix minimiza-tion 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 vec-tor minimization ..."

Abstract
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In this paper we study general Schatten-p quasi-norm (SPQN) regularized matrix minimiza-tion 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 vec-tor 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 mini-mization 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 minimiz-ers of the SPQN regularized matrix minimization problems. The iterative reweighted singular value minimization (IRSVM) methods are then proposed to solve these problems, whose sub-problems 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