A Deterministic Convergence Framework for Exact Non-Convex Phase Retrieval
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In this work, we analyze the non-convex framework of Wirtinger Flow (WF) 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.
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