Sparse Signal Recovery from Quadratic Measurements via Convex Programming

In this paper we consider a system of quadratic equations |〈zj ,x〉|2 = bj , j = 1, ...,m, where x ∈ R is unknown while normal random vectors zj ∈ R and quadratic measurements bj ∈ 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 logn ). On the other hand, we prove that k ≤ O(log n√m) is necessary for a class of naive convex relaxations to be exact.