The sparsity of LASSO-type minimizers

This note extends an attribute of the LASSO procedure to a whole class related procedures, including square-root LASSO, square LAD-LASSO, and instance generalized LASSO. Namely, under assumption that input matrix satisfies ℓ p -restricted isometry property (which in some sense is weaker than standard 2 assumption), it shown if vector comes from exact measurement sparse vector, then minimizer any such LASSO-type has sparsity comparable measured vector. The result remains valid presence moderate error when regularization parameter not too small.