Date of this Version
The Annals of Statistics
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.
The original and published work is available at: https://projecteuclid.org/euclid.aos/1473685274#abstract
Iterative adaptive thresholding, minimax rate, non-convex empirical risk, phase retrieval, sparse recovery, thresholded gradient method
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.