The Sparsity of LASSO-type Minimizers

TL;DR

This paper demonstrates that a broad class of LASSO-type procedures produce sparse solutions comparable to the true sparse vector under weaker restricted isometry conditions, even with some measurement error.

Contribution

It extends sparsity results to various LASSO variants under $ ext{ell}_p$-restricted isometry, broadening understanding of their sparsity properties.

Findings

LASSO-type minimizers are sparse under $ ext{ell}_p$-restricted isometry.

Results hold even with moderate measurement errors.

Sparsity of solutions is comparable to the true sparse vector.

Abstract

This note extends an attribute of the LASSO procedure to a whole class of related procedures, including square-root LASSO, square LASSO, LAD-LASSO, and an instance of generalized LASSO. Namely, under the assumption that the input matrix satisfies an $\ell_p$-restricted isometry property (which in some sense is weaker than the standard $\ell_2$-restricted isometry property assumption), it is shown that if the input vector comes from the exact measurement of a sparse vector, then the minimizer of any such LASSO-type procedure has sparsity comparable to the sparsity of the measured vector. The result remains valid in the presence of moderate measurement error when the regularization parameter is not too small.