噪声稀疏相位恢复的最优收敛率：基于阈值梯度下降法
Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow
AI总结：
本文研究噪声稀疏相位恢复问题，提出阈值梯度下降算法，在稀疏度范围内实现最优收敛率。
摘要：
本文考虑噪声稀疏相位恢复问题：从含噪声二次测量$y_j = (a_j' x )^2 + ε_j$恢复稀疏信号$x \in \mathbb{R}^p$。目标是理解稀疏性对估计精度的影响，并构造计算可行的估计器以达到最优收敛率。受Wirtinger Flow算法启发，提出新的阈值梯度下降算法，证明其在稀疏度范围内适应性地达到最小最大最优收敛率，当$a_j$为独立标准高斯随机向量且样本量足够大时。
英文摘要：
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 $x$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
