Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow
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This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal x ∈R^p from noisy quadratic measurements y_j = (a_j' x )^2 + ϵ_j, j=1, ..., m, with independent sub-exponential noise ϵ_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. Inspired by the Wirtinger Flow [12] proposed for noiseless and non-sparse 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.
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