Restricted Isometry Constants where $\ell^p$ sparse recovery can fail for $0 < p \leq 1

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) δ_2m 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 δ_2m arbitrarily close to one where L1 minimisation 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 minimisation recovers all m-sparse vectors for any matrix with δ_2m < 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 minimisation, Lp minimisation recovery failure is shown to be only slightly delayed in terms of the RIC values. Furthermore in this case the minimisation is nonconvex and it is important to consider the specific minimisation algorithm being used. It is shown that when Lp optimisation is attempted using an iterative reweighted L1 scheme, failure can still occur for δ_2m arbitrarily close to 1/√2.