Penalty Decomposition Methods for Rank Minimization

TL;DR

This paper introduces penalty decomposition methods for rank minimization problems, providing theoretical guarantees and demonstrating superior performance in matrix completion and correlation matrix tasks.

Contribution

It develops a novel penalty decomposition approach with block coordinate descent for rank minimization, including closed-form solutions for special cases.

Findings

Methods are comparable or superior to existing techniques

Theoretical convergence to first-order optimality

Effective in matrix completion and correlation matrix problems

Abstract

In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We first establish that a class of special rank minimization problems has closed-form solutions. Using this result, we then propose penalty decomposition methods for general rank minimization problems in which each subproblem is solved by a block coordinate descend method. Under some suitable assumptions, we show that any accumulation point of the sequence generated by the penalty decomposition methods satisfies the first-order optimality conditions of a nonlinear reformulation of the problems. Finally, we test the performance of our methods by applying them to the matrix completion and nearest low-rank correlation matrix problems. The computational results demonstrate that our methods are generally comparable or superior to the existing methods in terms of solution quality.