Two New Approaches to Compressed Sensing Exhibiting Both Robust Sparse Recovery and the Grouping Effect

TL;DR

This paper introduces the CLOT optimization method for compressed sensing, which combines the benefits of LASSO and Elastic Net by achieving robust sparse recovery and the grouping effect, unlike previous methods.

Contribution

The paper proposes the CLOT formulation, demonstrating its ability to recover sparse signals robustly and exhibit the grouping effect, improving upon existing methods like Elastic Net and SGL.

Findings

CLOT achieves robust sparse recovery in compressed sensing.

CLOT exhibits the grouping effect for highly correlated variables.

SGL also achieves robust recovery and grouping effect.

Abstract

In this paper we introduce a new optimization formulation for sparse regression and compressed sensing, called CLOT (Combined L-One and Two), wherein the regularizer is a convex combination of the $\ell_1$- and $\ell_2$-norms. This formulation differs from the Elastic Net (EN) formulation, in which the regularizer is a convex combination of the $\ell_1$- and $\ell_2$-norm squared. It is shown that, in the context of compressed sensing, the EN formulation does not achieve robust recovery of sparse vectors, whereas the new CLOT formulation achieves robust recovery. Also, like EN but unlike LASSO, the CLOT formulation achieves the grouping effect, wherein coefficients of highly correlated columns of the measurement (or design) matrix are assigned roughly comparable values. It is already known LASSO does not have the grouping effect. Therefore the CLOT formulation combines the best features of both LASSO (robust sparse recovery) and EN (grouping effect). The CLOT formulation is a special case of another one called SGL (Sparse Group LASSO) which was introduced into the literature previously, but without any analysis of either the grouping effect or robust sparse recovery. It is shown here that SGL achieves robust sparse recovery, and also achieves a version of the grouping effect in that coefficients of highly correlated columns belonging to the same group of the measurement (or design) matrix are assigned roughly comparable values.