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

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.

15 pp, to appear at IEEE Transactions on Signal Processing

References in corpus (4)

• A Selective Review of Group Selection in High-Dimensional Models
• Some sharp performance bounds for least squares regression with $L_1$ regularization
• The Linearized Inverse Problem in Multifrequency Electrical Impedance Tomography
• Error Bounds for Compressed Sensing Algorithms With Group Sparsity: A Unified Approach

Cited by in corpus (1)

• DC algorithms for a class of sparse group ${\ell }_0$ regularized optimization problems