Nearly optimal bounds for the global geometric landscape of phase retrieval

Abstract

Abstract The phase retrieval problem is concerned with recovering an unknown signal x ∈ C n from a set of magnitude-only measurements y j = ∣ ⟨ a j , x ⟩ ∣ , j = 1 , … , 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 et al (2018 Found. Comput. Math. 18 1131–98), in which the authors suggest that O ( n log n ) or even O(n) is enough to guarantee the favorable geometric property.