Non-convex compressed sensing with the sum-of-squares method

Abstract

We consider stable signal recovery in lq quasi-norm for 0 < q < 1. In this problem, given a measurement vector y = Ax for some unknown signal vector x ∈ ℝn and a known matrix A g ℝmxn, we want to recover z ∈ ℝn with ||x - z\\q = 0(\\x - x∗\\q) from a measurement vector, where x∗ is the s-sparse vector closest to x in lq quasi-norm. Although a small value of q is favorable for measuring the distance to sparse vectors, previous methods for q < 1 involve lq quasi-norm minimization which is computationally intractable. In this paper, we overcome this issue by using the sum-of-squares method, and give the first polynomial-time stable recovery scheme for a large class of matrices A in lq quasi-norm for any fixed constant 0 < q ≤ 1.