Error bound of critical points and KL property of exponent $1/2$ for squared F-norm regularized factorization

TL;DR

This paper analyzes the error bounds and KL property of squared F-norm regularized matrix factorization, providing theoretical guarantees for critical points and global minimizers in noisy low-rank matrix recovery.

Contribution

It establishes error bounds for critical points and proves the KL property of exponent 1/2 for the regularized loss function under noisy conditions.

Findings

Error bounds for non-strict critical points with rank constraints.

KL property of exponent 1/2 for the global minimizer set.

Validation through accelerated alternating minimization method.

Abstract

This paper is concerned with the squared F(robenius)-norm regularized factorization form for noisy low-rank matrix recovery problems. Under a suitable assumption on the restricted condition number of the Hessian for the loss function, we derive an error bound to the true matrix for the non-strict critical points with rank not more than that of the true matrix. Then, for the squared F-norm regularized factorized least squares loss function, under the noisy and full sample setting we establish its KL property of exponent $1/2$ on its global minimizer set, and under the noisy and partial sample setting achieve this property for a class of critical points. These theoretical findings are also confirmed by solving the squared F-norm regularized factorization problem with an accelerated alternating minimization method.