Statistics Papers

Document Type

Journal Article

Date of this Version

2016

Publication Source

The Annals of Statistics

Volume

44

Issue

5

Start Page

2221

Last Page

2251

DOI

10.1214/16-AOS1443

Abstract

This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal x ∈ ℝp from noisy quadratic measurements yj = (ajx)2+εj, j=1,…,m, with independent sub-exponential noise εj. The goals are to understand the effect of the sparsity of 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 aj’s are independent standard Gaussian random vectors, provided that the sample size is sufficiently large compared to the sparsity of x.

Copyright/Permission Statement

The original and published work is available at: https://projecteuclid.org/euclid.aos/1473685274#abstract

Keywords

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

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Date Posted: 27 November 2017

This document has been peer reviewed.