Provable sample-efficient sparse phase retrieval initialized by truncated power method

Abstract We study the sparse phase retrieval problem, recovering an s -sparse length- n signal from m magnitude-only measurements. Two-stage non-convex approaches have drawn much attention in recent studies. Despite non-convexity, many two-stage algorithms provably converge to underlying solution linearly when appropriately initialized. However, terms of sample complexity, bottleneck those with Gaussian random measurements often comes initialization stage. Although refinement stage usually needs only m = Ω ( s log n stretchy="false">) measurements, widely used spectral requires \Omega(s^2\log Ω 2 form="prefix">log produce a desired initial guess, which causes total complexity order-wisely more than necessary. To reduce number we propose truncated power method replace for algorithms. prove that \Omega(\bar{s} s\log ˉ where $\bar{s}$?> is stable sparsity signal, are sufficient guess. When contains very few significant components, proposed algorithm and optimal. Numerical experiments illustrate sample-efficient state-of-the-art