Two New Approaches to Compressed Sensing Exhibiting Both Robust Sparse Recovery and the 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).