Nearly optimal bounds for the global geometric landscape of phase retrieval

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.

Automated summary

The phase retrieval problem involves recovering an unknown signal x from magnitude-only measurements. A natural least squares formulation can efficiently solve this problem, even with random initialization, despite the non-convexity of the loss function. The paper shows that m = O(n log n) Gaussian random measurements are sufficient to ensure the favorable geometric property of a commonly used estimator with high probability, progressing towards answering an open problem in the field.