Restricted isometry constants where lpsparse recovery can fail for 0 < p <= 1

Abstract

This paper investigates conditions under which the solution of an underdetermined linear system with minimal lP norm, 0 < p ≤ 1, is guaranteed to be also the sparsest one. Matrices are constructed with restricted isometry constants (RIC) δ2m arbitrarily close to 1/ √2 ≅ 0.707 where sparse recovery with p = 1 fails for at least one m-sparse vector, as well as matrices with δ2m arbitrarily close to one where l1 minimization succeeds for any m-sparse vector. This highlights the pessimism of sparse recovery prediction based on the RIC, and indicates that there is limited room for improving over the best known positive results of Foucart and Lai, which guarantee that l1 minimization recovers all m-sparse vedors for any matrix with δ2m < 2(3 - √2)/7 ≅ 0.4531. These constructions are a by-product of tight conditions for lp recovery (0 ≤ p ≤ 1) with matrices of unit spectral norm, which are expressed in terms of the minimal singular values of 2m-column submatrices. Compared to l1 minimization, lp minimization recovery failure is shown to be only slightly delayed in terms of the RIC values. Furthermore in this case the minimization is nonconvex and it is important to consider the specific minimization algorithm being used. It is shown that when lp optimization is attempted using an iterative reweighted l1 scheme, failure can still occur for δ2m arbitrarily close to 1/ √2.