Phase retrieval with random Gaussian sensing vectors by alternating projections

TL;DR

This paper analyzes the effectiveness of alternating projections for phase retrieval with Gaussian sensing vectors, showing success under certain measurement regimes and highlighting the role of initialization and stagnation points.

Contribution

It demonstrates that alternating projections can reliably solve phase retrieval with Gaussian vectors when measurements are sufficiently numerous and discusses regimes where initialization is unnecessary.

Findings

Success with high probability when m ≥ Cn with proper initialization

Disappearance of stagnation points when m is of order n^2

Potential for success in the regime m=O(n) with small attraction basins

Abstract

We consider a phase retrieval problem, where we want to reconstruct a $n$-dimensional vector from its phaseless scalar products with $m$ sensing vectors. We assume the sensing vectors to be independently sampled from complex normal distributions. We propose to solve this problem with the classical non-convex method of alternating projections. We show that, when $m\geq Cn$ for $C$ large enough, alternating projections succeed with high probability, provided that they are carefully initialized. We also show that there is a regime in which the stagnation points of the alternating projections method disappear, and the initialization procedure becomes useless. However, in this regime, $m$ has to be of the order of $n^2$. Finally, we conjecture from our numerical experiments that, in the regime $m=O(n)$, there are stagnation points, but the size of their attraction basin is small if $m/n$ is large enough, so alternating projections can succeed with probability close to $1$ even with no special initialization.