Approximate methods for phase retrieval via gauge duality

TL;DR

This paper introduces an efficient approach for phase retrieval by combining gauge duality with approximate spectral computations and gradient descent, enabling low-cost recovery of low-rank matrices.

Contribution

It proposes a novel method that uses approximate spectral computations in gauge duality to initialize a gradient descent scheme for phase retrieval.

Findings

Consistent recovery on small problems

Low computational cost

Effective initialization for nonconvex optimization

Abstract

We consider the problem of finding a low rank symmetric matrix satisfying a system of linear equations, as appears in phase retrieval. In particular, we solve the gauge dual formulation, but use a fast approximation of the spectral computations to achieve a noisy solution estimate. This estimate is then used as the initialization of an alternating gradient descent scheme over a nonconvex rank-1 matrix factorization formulation. Numerical results on small problems show consistent recovery, with very low computational cost.