Sparse Signal Recovery from Quadratic Measurements via Convex Programming

TL;DR

This paper demonstrates that sparse signal recovery from quadratic measurements can be achieved via convex programming under certain sparsity conditions, with theoretical guarantees on success and limitations of naive relaxations.

Contribution

It establishes conditions under which convex optimization can recover sparse signals from quadratic measurements and identifies the bounds where naive relaxations fail.

Findings

Recovery is possible if sparsity k <= O((m/log n)^(1/2)).

Naive convex relaxations require k <= O(log n * (m)^(1/2)) for exactness.

High probability guarantees are provided for the proposed method.

Abstract

In this paper we consider a system of quadratic equations |<z_j, x>|^2 = b_j, j = 1, ..., m, where x in R^n is unknown while normal random vectors z_j in R_n and quadratic measurements b_j 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.