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 = (a′jx)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
Recommended Citation
Cai, T., Li, X., & Ma, Z. (2016). Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow. The Annals of Statistics, 44 (5), 2221-2251. http://dx.doi.org/10.1214/16-AOS1443
Date Posted: 27 November 2017
This document has been peer reviewed.