Construction of optimal spectral methods in phase retrieval

arXiv2012.04524MaRDI QIDQ6355586

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Publication date: 8 December 2020

Abstract: We consider the phase retrieval problem, in which the observer wishes to recover a $n$-dimensional real or complex signal $mathbf{X}^{s}tar$ from the (possibly noisy) observation of $|mathbfPhimathbf{X}^{s}tar|$, in which $mathbfPhi$ is a matrix of size $mimesn$. We consider a emph{high-dimensional} setting where $n,moinfty$ with $m/n=mathcalO(1)$, and a large class of (possibly correlated) random matrices $mathbfPhi$ and observation channels. Spectral methods are a powerful tool to obtain approximate observations of the signal $mathbf{X}^{s}tar$ which can be then used as initialization for a subsequent algorithm, at a low computational cost. In this paper, we extend and unify previous results and approaches on spectral methods for the phase retrieval problem. More precisely, we combine the linearization of message-passing algorithms and the analysis of the emph{Bethe Hessian}, a classical tool of statistical physics. Using this toolbox, we show how to derive optimal spectral methods for arbitrary channel noise and right-unitarily invariant matrix $mathbfPhi$, in an automated manner (i.e. with no optimization over any hyperparameter or preprocessing function).

Has companion code repository: https://github.com/AnMaillard/Optimal_Spectral_Methods_PR

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