On Lp minimisation, instance optimality, and restricted isometry constants for sparse approximation
Résumé
We extend recent results regarding the restricted isometry constants (RIC) and sparse recovery using ℓp minimisation. Here we consider the case of the sparse approximation of compressible rather than exactly sparse signals. We begin by showing that the robust null space property used in [3] characterises the robustness of the estimation of compressible signals by ℓp mimisation for all ℓp norms, 0 < p ≤ 1 and not just ℓ1, in the sense of instance optimality. Furthermore we show that this characterisation is in fact sharp in a particular sense. We are then able to simply extend the results in [6] to show that matrices that fail to exhibit good constants for instance optimality can be found with relatively small RICs.
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