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2013

Minimax rate of convergence and the performance of ERM in phase recovery

Lecué, Guillaume, Mendelson, Shahar

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

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.

Minimax rate of convergence and the performance of ERM in phase recovery · Around

Built on

L. Birgé,
1983
Earlier work this paper cites.
The widths of a Euclidean ball
A. Yu. Garnaev and E. D. Gluskin · 1984
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V.D. Milman, G. Schechtman,
1986
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Subspaces of small codimension of finite-dimensional Banach spaces
Alain Pajor and Nicole Tomczak-Jaegermann · 1986
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M. Ledoux, M. Talagrand,
1991
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