Sparse Signal Recovery from Quadratic Measurements via Convex Programming

In this paper we consider a system of quadratic equations |<z_j, x>|² = bj, j = 1, ..., m, where x in Rⁿ is unknown while normal random vectors zj in Rn and quadratic measurements bj in R are known. The system is assumed to be underdetermined, i.e., m < n. We prove that if there exists a sparse solution x, i.e., at most k components of x are non-zero, then by solving a convex optimization program, we can solve for x up to a multiplicative constant with high probability, provided that k <= O((m/log n)^1/2). On the other hand, we prove that k <= O(log n (m)^1/2) is necessary for a class of naive convex relaxations to be exact.