Reconstruction of sparse signals from highly corrupted measurements by nonconvex minimization

We propose a method for signal recovery in compressed sensing when measurements can be highly corrupted. It is based on ℓ
_p
minimization for 0 <; p ≤ 1. Since it was shown that ℓ
_p
minimization performs better than ℓ
₁
minimization when there are no large errors, the proposed approach is a natural extension to compressed sensing with corruptions. We provide a theoretical justification of this idea, based on analogous reasoning as in the case when measurements are not corrupted by large errors. Better performance of the proposed approach compared to ℓ
₁
minimization is illustrated in numerical experiments.