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

We study the performance of Empirical Risk Minimization in both noisy and noiseless phase retrieval problems, indexed by subsets of R and relative to subgaussian sampling; that is, when the given data is yi = 〈 ai, x0 〉2 +wi for a subgaussian random vector a, independent subgaussian noise w and a fixed but unknown x0 that belongs to a given T ⊂ R. We show that ERM performed in T produces x̂ whose Euclidean distance to either x0 or −x0 depends on the gaussian mean-width of T and on the signal-to-noise ratio of the problem. The bound coincides with the one for linear regression when ‖x0‖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 R and the unit ball in `1 .