KL property of exponent $1/2$ of $\ell_{2,0}$-norm and DC regularized factorizations for low-rank matrix recovery

TL;DR

This paper investigates the KL property of a specific exponent in low-rank matrix recovery problems with regularization, providing theoretical guarantees and validating them through an optimization algorithm.

Contribution

It establishes the KL property of exponent 1/2 for a regularized factored loss function in low-rank matrix recovery, under certain conditions.

Findings

Proves KL property of exponent 1/2 for the regularized problem

Validates theoretical results with a proximal linearized alternating minimization method

Provides conditions under which the KL property holds

Abstract

This paper is concerned with the factorization form of the rank regularized loss minimization problem. To cater for the scenario in which only a coarse estimation is available for the rank of the true matrix, an $\ell_{2,0}$-norm regularized term is added to the factored loss function to reduce the rank adaptively; and account for the ambiguities in the factorization, a balanced term is then introduced. For the least squares loss, under a restricted condition number assumption on the sampling operator, we establish the KL property of exponent $1/2$ of the nonsmooth factored composite function and its equivalent DC reformulations in the set of their global minimizers. We also confirm the theoretical findings by applying a proximal linearized alternating minimization method to the regularized factorizations.