Nearly optimal bounds for the global geometric landscape of phase retrieval

TL;DR

This paper proves that with about O(n log n) Gaussian measurements, the phase retrieval loss landscape is benign, with all local minima being global and saddle points having negative curvature, aiding efficient recovery.

Contribution

It establishes nearly optimal bounds on the number of measurements needed for a benign geometric landscape in phase retrieval, advancing understanding of its non-convex optimization.

Findings

O(n log n) measurements suffice for benign landscape

All local minima are global with high probability

Negative curvature exists around saddle points

Abstract

The phase retrieval problem is concerned with recovering an unknown signal $\bf{x} \in \mathbb{R}^n$ from a set of magnitude-only measurements $y_j=|\langle \bf{a}_j,\bf{x} \rangle|, \; j=1,\ldots,m$. A natural least squares formulation can be used to solve this problem efficiently even with random initialization, despite its non-convexity of the loss function. One way to explain this surprising phenomenon is the benign geometric landscape: (1) all local minimizers are global; and (2) the objective function has a negative curvature around each saddle point and local maximizer. In this paper, we show that $m=O(n \log n)$ Gaussian random measurements are sufficient to guarantee the loss function of a commonly used estimator has such benign geometric landscape with high probability. This is a step toward answering the open problem given by Sun-Qu-Wright, in which the authors suggest that $O(n \log n)$ or even $O(n)$ is enough to guarantee the favorable geometric property.