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October 2016 Optimal rates of convergence for noisy sparse phase retrieval via thresholded Wirtinger flow
T. Tony Cai, Xiaodong Li, Zongming Ma
Ann. Statist. 44(5): 2221-2251 (October 2016). DOI: 10.1214/16-AOS1443

Abstract

This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal $\mathbf{x}\in\mathbb{R}^{p}$ from noisy quadratic measurements $y_{j}=(\mathbf{a}_{j}'\mathbf{x})^{2}+\varepsilon_{j}$, $j=1,\ldots,m$, with independent sub-exponential noise $\varepsilon_{j}$. The goals are to understand the effect of the sparsity of $\mathbf{x}$ on the estimation precision and to construct a computationally feasible estimator to achieve the optimal rates adaptively. Inspired by the Wirtinger Flow [IEEE Trans. Inform. Theory 61 (2015) 1985–2007] proposed for non-sparse and noiseless phase retrieval, a novel thresholded gradient descent algorithm is proposed and it is shown to adaptively achieve the minimax optimal rates of convergence over a wide range of sparsity levels when the $\mathbf{a}_{j}$’s are independent standard Gaussian random vectors, provided that the sample size is sufficiently large compared to the sparsity of $\mathbf{x}$.

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T. Tony Cai. Xiaodong Li. Zongming Ma. "Optimal rates of convergence for noisy sparse phase retrieval via thresholded Wirtinger flow." Ann. Statist. 44 (5) 2221 - 2251, October 2016. https://doi.org/10.1214/16-AOS1443

Information

Received: 1 June 2015; Revised: 1 January 2016; Published: October 2016
First available in Project Euclid: 12 September 2016

zbMATH: 1349.62019
MathSciNet: MR3546449
Digital Object Identifier: 10.1214/16-AOS1443

Subjects:
Primary: 62C20
Secondary: 62P35

Keywords: Iterative adaptive thresholding , Minimax rate , non-convex empirical risk , phase retrieval , sparse recovery , thresholded gradient method

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 5 • October 2016
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