Papers › Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow
Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow
T. Tony Cai, Xiao-Dong Li, Zongming Ma
This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal x ∈ℝᵖ from noisy quadratic measurements yⱼ = (aⱼ′ x )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. 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 aⱼ'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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