Factorized low-rank matrix recovery problem, Schatten-$q$ quasi-norm, Error bound for critical point, Kurdyka-Łojasiewicz property, Inexact proximal alternating linearized minimization

Abstract

The Schatten-$q$ quasi-norm is a widely used nonconvex rank surrogate and matrix factorization is an effective approach to reduce computational cost. In this paper, we consider the equivalent group-sparse factorized reformulation of Schatten-$q$ norm regularized low-rank matrix recovery problem. Though this factorized model exhibits favorable performance, two issues remain: (i) the error bound of critical points is unexplored; (ii) the proximal operator of $\|\cdot\|_2^q$ lacks a closed-form solution for general $q$, limiting algorithms to adopt fixed $q$ like $1/2$ or $2/3$. This paper addresses both issues. We investigate the properties of critical points for the factorized problem and show that, compared to nuclear norm, the Schatten-$q$ norm implicitly endows critical points with column orthogonality. From this insight, we introduce the notion of S-critical points under mild conditions that ensure column orthogonality with easily operable criterion for identifying. We show that global optimal points must be S-critical points and we derive an error bound between S-critical points and the true matrix. We further present an inexact proximal alternating linearized minimization method for the factorized problem, along with practically computable inexact proximal operator for $\|\cdot\|_2^q$ and criteria to find solutions satisfying inexactness conditions, and we establish the whole sequence convergence and a convergence rate guarantee under Kurdyka--Łojasiewicz condition. Moreover, we prove that the factorized model with least-squares loss has KL exponent $1/2$ at S-critical points, then the iteration converges linearly under suitable condition. Extensive numerical experiments validate the effectiveness of our algorithm and confirm the theoretical properties of the factorized model.