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 $\mathbf{x}$.
Citation
T. Tony Cai. Xiaodong Li. Zongming Ma. "Optimal rates of convergence for noisy sparse phase retrieval via thresholded Wirtinger flow." Ann. Statist. 44 (5) 2221 - 2251, October 2016. https://doi.org/10.1214/16-AOS1443
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