OPTIMAL RATES OF CONVERGENCE FOR NOISY SPARSE PHASE RETRIEVAL VIA THRESHOLDED WIRTINGER FLOW

摘要

This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal x is an element of R-P from noisy quadratic measurements y(j) = (a(j)'x)(2) + epsilon(j), j = 1,... m, with independent sub-exponential noise epsilon(j). 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 a(j)'s are independent standard Gaussian random vectors, provided that the sample size is sufficiently large compared to the sparsity of x.