Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow
This work addresses the challenge of signal recovery in noisy, sparse settings for applications like imaging and sensing, representing an incremental improvement by adapting an existing method to incorporate sparsity and noise.
The paper tackles the noisy sparse phase retrieval problem by recovering a sparse signal from noisy quadratic measurements, proposing a thresholded gradient descent algorithm that achieves minimax optimal convergence rates across various sparsity levels with sufficient sample size.
This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal $x \in \mathbb{R}^p$ from noisy quadratic measurements $y_j = (a_j' x )^2 + ε_j$, $j=1, \ldots, m$, 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 $x$.