SPARSE SIGNAL RECOVERY FROM QUADRATIC MEASUREMENTS VIA CONVEX PROGRAMMING

In this paper we consider a system of quadratic equations vertical bar < z(j), x > |(2) = b(j), j - 1,..., m, where x is an element of R-n is unknown while normal random vectors z is an element of R-n and quadratic measurements b I 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 nonzero, then by solving a convex optimization program, we can solve for x up to a multiplicative constant with high probability, provided that k <= O root m/log n). On the other hand, we prove that k <= O( log n root m) is necessary for a class of natural convex relaxations to be exact.