## Electronic Journal of Probability

### 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 $\ell_1^n$.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 57, 29 pp.

Dates
Accepted: 29 May 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067163

Digital Object Identifier
doi:10.1214/EJP.v20-3525

Mathematical Reviews number (MathSciNet)
MR3354617

Zentralblatt MATH identifier
1318.62178

Subjects
Primary: 62G99: None of the above, but in this section

Rights

#### Citation

Lecué, Guillaume; Mendelson, Shahar. Minimax rate of convergence and the performance of empirical risk minimization in phase recovery. Electron. J. Probab. 20 (2015), paper no. 57, 29 pp. doi:10.1214/EJP.v20-3525. https://projecteuclid.org/euclid.ejp/1465067163

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