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

Abstract

We study the performance of Empirical Risk Minimization in both noisy and noiseless phase retrieval problems, indexed by subsets of $\mathbb{R}^n$ and relative to subgaussian sampling; that is, when the given data is $y_i=\left a_i,x_0\right^2+w_i$ for a subgaussian random vector $a$, independent noise $w$ and a fixed but unknown $x_0$ that belongs to a given subset of $\mathbb{R}^n$.

We show that ERM produces $\hat{x}$ whose Euclidean distance to either $x_0$ or $-x_0$ depends on the gaussian mean-width of the indexing set and on the signal-to-noise ratio of the problem. The bound coincides with the one for linear regression when $\|x_0\|_2$ is of the order of a constant. In addition, we obtain a minimax lower bound for the problem and identify sets for which ERM is a minimax procedure. As examples, we study the class of $d$-sparse vectors in $\mathbb{R}^n$ and the unit ball in $\ell_1^n$.