A Deterministic Theory for Exact Non-Convex Phase Retrieval

Abstract

In this work, we analyze the non-convex framework of Wirtinger Flow (WF) [1] for phase retrieval and identify a novel sufficient condition for universal exact recovery through the lens of low rank matrix recovery theory. Via a perspective in the lifted domain, we establish that the convergence of WF to a true solution is geometrically implied under a condition on the lifted forward model which relates to the concentration of the spectral matrix around its expectation given that the bound is sufficiently tight. As a result, a deterministic relationship between accuracy of spectral initialization and the validity of the regularity condition is derived, and a convergence rate that solely depends on the concentration bound is obtained. Notably, the developed framework addresses a theoretical gap in non-convex optimization literature on solving quadratic systems of equations with the convergence arguments that are deterministic. Finally, we quantify a lower bound on the signal-to-noise ratio such that theoretical guarantees are valid using the spectral initialization even in the absence of pre-processing or sample truncation.

An Optimal Statistical and Computational Framework for Generalized Tensor Estimation

“…Remark 3.1 (Interpretation of the RCG Condition) The RCG condition is similar to the "regularity condition" appearing in recent nonconvex optimization literature (Chen and Candes, 2015;Candes et al, 2015;Chi et al, 2019;Yonel and Yazici, 2020):…”