Regularization parameter selection for low rank matrix recovery

TL;DR

This paper introduces a novel rule for selecting the regularization parameter in nuclear norm regularized matrix recovery, improving efficiency by leveraging duality theory and feasible points, with practical formulas for specific loss functions.

Contribution

It develops a new parameter selection rule for NRM based on duality, filling a gap in matrix case regularization parameter choice methods.

Findings

The rule simplifies choosing regularization parameters for NRM.

Numerical results show the rule effectively shrinks the parameter interval.

Formulas are provided for least squares and Huber functions.

Abstract

Low rank matrix recovery is the focus of many applications, but it is a NP-hard problem. A popular way to deal with this problem is to solve its convex relaxation, the nuclear norm regularized minimization problem (NRM), which includes LASSO as a special case. There are some regularization parameter selection results for LASSO in vector case, such as screening rules, which improve the efficiency of the algorithms. However, there are no corresponding parameter selection results for NRM in matrix case. In this paper, we build up a novel rule to choose the regularization parameter for NRM under the help of duality theory. This rule claims that the regularization parameter can be easily chosen by feasible points of NRM and its dual problem, when the rank of the desired solution is no more than a given constant. In particular, we apply this idea to NRM with least square and Huber functions, and establish the easily calculated formula of regularization parameters. Finally, we report numerical results on some signal shapes, which state that our proposed rule shrinks the interval of the regularization parameter efficiently.