Minimax rate of convergence and the performance of empirical risk minimization in phase retrieval

Abstract

We study the performance of Empirical Risk Minimization in both noisy and noiseless phase retrieval problems, indexed by subsets of ℝ n and relative to subgaussian sampling; that is, when the given data is y i = 〈a i; x 0 〉 2 + w i for a subgaussian random vector a, independent subgaussian noise w and a fixed but unknown x 0 that belongs to a given T ⊂ ℝ n. We show that ERM performed in T produces ẍ whose Euclidean distance to either x 0 or x 0 depends on the gaussian mean-width of T and on the signal-to-noise atio of the problem. The bound coincides with the one for linear regression when ǁx 0 ǁ 2 is of the order of a constant. In addition, we obtain a sharp lower bound for the phase retrieval problem. As examples, we study the class of d-sparse vectors in ℝn and the unit ball in ∫ n 1.