Low-rank matrix recovery via nonconvex optimization methods with application to errors-in-variables matrix regression

TL;DR

This paper introduces a nonconvex regularized approach for low-rank matrix recovery, providing tighter recovery bounds than convex methods under certain singular value conditions, and applies it to errors-in-variables matrix regression.

Contribution

It develops a nonconvex optimization framework with theoretical recovery guarantees and demonstrates its advantages in errors-in-variables matrix regression scenarios.

Findings

Recovery bounds are tighter than convex methods for large singular values.

The nonconvex method performs well with additive noise and missing data.

Probabilistic analysis confirms the advantages of the nonconvex approach.

Abstract

We consider the nonconvex regularized method for low-rank matrix recovery. Under the assumption on the singular values of the parameter matrix, we provide the recovery bound for any stationary point of the nonconvex method by virtue of regularity conditions on the nonconvex loss function and the regularizer. This recovery bound can be much tighter than that of the convex nuclear norm regularized method when some of the singular values are larger than a threshold defined by the nonconvex regularizer. In addition, we consider the errors-in-variables matrix regression as an application of the nonconvex optimization method. Probabilistic consequences and the advantage of the nonoconvex method are demonstrated through verifying the regularity conditions for specific models with additive noise and missing data.