Two new approaches to compressed sensing exhibiting both robust sparse recovery and the grouping effect

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 ℓ₁- and ℓ₂-norms. This formulation differs from the Elastic Net (EN) formulation, in which the regularizer is a convex combination of the ℓ₁- and ℓ₂-norm squared. This seemingly simple modification has fairly significant consequences. In particular, it is shown in this paper that the EN formulation does not achieve robust recovery of sparse vectors in the context of compressed sensing, whereas the new CLOT formulation does so. 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 noteworthy that LASSO does not have the grouping effect and EN (as shown here) does not achieve robust sparse recovery. 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 of the measurement (or design) matrix are assigned roughly comparable values, if the columns belong to the same group.