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In this paper we introduce a new optimization formulation for sparseregression 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, whereasthe new CLOT formulation achieves robust recovery. Also, like EN but unlikeLASSO, the CLOT formulation achieves the grouping effect, wherein coefficientsof highly correlated columns of the measurement (or design) matrix are assignedroughly comparable values. It is already known LASSO does not have the groupingeffect. 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 (SparseGroup LASSO) which was introduced into the literature previously, but withoutany analysis of either the grouping effect or robust sparse recovery. It isshown here that SGL achieves robust sparse recovery, and also achieves aversion of the grouping effect in that coefficients of highly correlatedcolumns belonging to the same group of the measurement (or design) matrix areassigned roughly comparable values.

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