## The Annals of Statistics

### Optimal rates of convergence for noisy sparse phase retrieval via thresholded Wirtinger flow

#### 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 $\mathbf{x}$.

#### Article information

Source
Ann. Statist. Volume 44, Number 5 (2016), 2221-2251.

Dates
Revised: January 2016
First available in Project Euclid: 12 September 2016

https://projecteuclid.org/euclid.aos/1473685274

Digital Object Identifier
doi:10.1214/16-AOS1443

Mathematical Reviews number (MathSciNet)
MR3546449

Zentralblatt MATH identifier
1349.62019

Subjects
Primary: 62C20: Minimax procedures
Secondary: 62P35: Applications to physics

#### Citation

Cai, T. Tony; Li, Xiaodong; Ma, Zongming. Optimal rates of convergence for noisy sparse phase retrieval via thresholded Wirtinger flow. Ann. Statist. 44 (2016), no. 5, 2221--2251. doi:10.1214/16-AOS1443. https://projecteuclid.org/euclid.aos/1473685274.

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