Support recovery with sparsely sampled free random matrices

Consider a Bernoulli-Gaussian complex n-vector whose components are X i B i , with B i ∼Bernoulli-q and X i ∼ CN(0; σ2), iid across i and mutually independent. This random q-sparse vector is multiplied by a random matrix U, and a randomly chosen subset of the components of average size np, p ∈ [0; 1], of the resulting vector is then observed in additive Gaussian noise. We extend the scope of conventional noisy compressive sampling models where U is typically the identity or a matrix with iid components, to allow U that satisfies a certain freeness condition, which encompasses Haar matrices and other unitarily invariant matrices. We use the replica method and the decoupling principle of Guo and Verdu, as well as a number of information theoretic bounds, to study the input-output mutual information and the support recovery error rate as n → ∞.