Group Sparse Recovery via the $\ell^0(\ell^2)$ Penalty: Theory and Algorithm

TL;DR

This paper introduces a novel $ ext{l}^0( ext{l}^2)$ regularized approach for group sparse recovery that effectively handles strong inner-group correlations and provides a fast, convergent algorithm with proven theoretical guarantees.

Contribution

The paper proposes a new $ ext{l}^0( ext{l}^2)$ penalty model for group sparse recovery, along with an efficient primal dual active set algorithm with finite-step convergence.

Findings

The method effectively handles strong inner-group correlations.

The algorithm converges in finite steps with proven global convergence.

Numerical experiments show high efficiency and accuracy, outperforming some existing methods.

Abstract

In this work we propose and analyze a novel approach for group sparse recovery. It is based on regularized least squares with an $\ell^0(\ell^2)$ penalty, which penalizes the number of nonzero groups. One distinct feature of the approach is that it has the built-in decorrelation mechanism within each group, and thus can handle challenging strong inner-group correlation. We provide a complete analysis of the regularized model, e.g., existence of a global minimizer, invariance property, support recovery, and properties of block coordinatewise minimizers. Further, the regularized problem admits an efficient primal dual active set algorithm with a provable finite-step global convergence. At each iteration, it involves solving a least-squares problem on the active set only, and exhibits a fast local convergence, which makes the method extremely efficient for recovering group sparse signals. Extensive numerical experiments are presented to illustrate salient features of the model and the efficiency and accuracy of the algorithm. A comparative study indicates its competitiveness with existing approaches.