Deterministic phase guarantees for robust recovery in incoherent dictionaries

This paper presents a relaxation of an assumption usually imposed in the recovery of sparse vectors with random support in pairs of orthonormal bases or incoherent dictionaries by basis pursuit. The assumption requires the phases of the entries of the sparse vector to be chosen randomly in [0, 2π). This paper provides probabilistic recovery guarantees for deterministic phases. We prove that, if a phase pattern is fixed, then a sparse vector with random support and corresponding phases can be recovered with high probability. As a result, the phases can take any distribution and can be dependent, as long as they are independent of the support. Furthermore, this improvement does not come at the expense of the maximum recoverable sparsity.