The Restricted Isometry Property and ` sparse recovery failure

This paper considers conditions based on the restricted isometry constant (RIC) under which the solution of an underdetermined linear system with minimal ` norm, 0 < p ≤ 1, is guaranteed to be also the sparsest one. Specifically matrices are identified that have RIC, δ2m, arbitrarily close to 1/ √ 2 ≈ 0.707 where sparse recovery with p = 1 fails for at least one m-sparse vector. This indicates that there is limited room for improvement over the best known positive results of Foucart and Lai, which guarantee that `-minimisation recovers all m-sparse vectors for any matrix with δ2m < 2(3− √ 2)/7 ≈ 0.4531. We also present results that show, compared to ` minimisation, ` minimisation recovery failure is only slightly delayed in terms of the RIC values. Furthermore when ` optimisation is attempted using an iterative reweighted ` scheme, failure can still occur for δ2m arbitrarily close to 1/ √ 2.