Penalty Decomposition Methods for $L0$-Norm Minimization

TL;DR

This paper introduces a penalty decomposition method for l0-norm minimization problems, reformulating them as rank minimization problems and demonstrating improved performance in compressed sensing and sparse regression tasks.

Contribution

The paper develops a vector-operation-based penalty decomposition method for l0-norm minimization, extending it to objective functions and showing superior results over existing methods.

Findings

Method outperforms existing approaches in solution quality.

Method demonstrates faster convergence in experiments.

Applicable to compressed sensing and sparse regression.

Abstract

In this paper we consider general l0-norm minimization problems, that is, the problems with l0-norm appearing in either objective function or constraint. In particular, we first reformulate the l0-norm constrained problem as an equivalent rank minimization problem and then apply the penalty decomposition (PD) method proposed in [33] to solve the latter problem. By utilizing the special structures, we then transform all matrix operations of this method to vector operations and obtain a PD method that only involves vector operations. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the PD method satisfies a first-order optimality condition that is generally stronger than one natural optimality condition. We further extend the PD method to solve the problem with the l0-norm appearing in objective function. Finally, we test the performance of our PD methods by applying them to compressed sensing, sparse logistic regression and sparse inverse covariance selection. The computational results demonstrate that our methods generally outperform the existing methods in terms of solution quality and/or speed.