The Annals of Statistics · 2016 · 208 citations · 45 references
Natural SciencesComputer ScienceMathematicsInformation RetrievalInformation TheoryOptimizationConvex OptimizationDigital Signal ProcessingSignal ProcessingData SparsityGradient DescentMinimax RatesOptimal Convergence RatesPhase RetrievalSparse ReconstructionSparse Signal ProcessingSparsity Problem
This paper studies the noisy sparse phase retrieval problem of recovering a sparse signal from noisy quadratic measurements, a setting where the signal is observed through squared linear projections corrupted by sub‑exponential noise, and it builds on the Wirtinger Flow framework. The goal is to understand how sparsity affects estimation precision and to construct a computationally feasible estimator that achieves optimal rates adaptively. The authors propose a thresholded gradient descent algorithm based on Wirtinger Flow for sparse phase retrieval with noisy quadratic measurements. The algorithm adaptively achieves minimax optimal convergence rates across a wide range of sparsity levels when the measurement vectors are independent standard Gaussian and the sample size exceeds a threshold proportional to sparsity. No additional information provided.
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 $\mathbf{x}$.
45