Stable Phase Retrieval with Mirror Descent

TL;DR

This paper extends mirror descent methods to noisy phase retrieval, proving stability and convergence guarantees, and demonstrating their efficiency through theoretical analysis and numerical experiments.

Contribution

It introduces a stable mirror descent algorithm for noisy phase retrieval with convergence guarantees and improved initialization strategies.

Findings

Mirror descent converges to critical points in noiseless cases.

Under noise, the method remains stable and converges to near the true vector.

Numerical results confirm computational and statistical efficiency.

Abstract

In this paper, we aim to reconstruct an n-dimensional real vector from m phaseless measurements corrupted by an additive noise. We extend the noiseless framework developed in [15], based on mirror descent (or Bregman gradient descent), to deal with noisy measurements and prove that the procedure is stable to (small enough) additive noise. In the deterministic case, we show that mirror descent converges to a critical point of the phase retrieval problem, and if the algorithm is well initialized and the noise is small enough, the critical point is near the true vector up to a global sign change. When the measurements are i.i.d Gaussian and the signal-to-noise ratio is large enough, we provide global convergence guarantees that ensure that with high probability, mirror descent converges to a global minimizer near the true vector (up to a global sign change), as soon as the number of measurements m is large enough. The sample complexity bound can be improved if a spectral method is used to provide a good initial guess. We complement our theoretical study with several numerical results showing that mirror descent is both a computationally and statistically efficient scheme to solve the phase retrieval problem.