Nearly optimal bounds for the global geometric landscape of phase retrieval

Abstract

The phase retrieval problem is concerned with recovering an unknown signal from a set of magnitude-only measurements . 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 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 et al (2018 Found. Comput. Math. 18 1131–98), in which the authors suggest that or even O(n) is enough to guarantee the favorable geometric property.