Sparse phase retrieval via group-sparse optimization

TL;DR

This paper introduces a group-sparse optimization approach for sparse phase retrieval, providing theoretical guarantees for exact recovery and stability in both real and complex cases.

Contribution

It reformulates sparse phase retrieval as a group-sparse optimization problem and analyzes its convex relaxation with provable recovery guarantees.

Findings

Convex relaxation via block l1-norm achieves exact recovery.

Provides stability results under measurement noise.

Discusses invariance properties for real vectors.

Abstract

This paper deals with sparse phase retrieval, i.e., the problem of estimating a vector from quadratic measurements under the assumption that few components are nonzero. In particular, we consider the problem of finding the sparsest vector consistent with the measurements and reformulate it as a group-sparse optimization problem with linear constraints. Then, we analyze the convex relaxation of the latter based on the minimization of a block l1-norm and show various exact recovery and stability results in the real and complex cases. Invariance to circular shifts and reflections are also discussed for real vectors measured via complex matrices.