Nearly optimal bounds for the global geometric landscape of phase retrieval
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The phase retrieval problem is concerned with recovering an unknown signal x∈ℝ^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-Qu-Wright, in which the authors suggest that O(n log n) or even O(n) is enough to guarantee the favorable geometric property.
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