Construction of optimal spectral methods in phase retrieval
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We consider the phase retrieval problem, in which the observer wishes to recover a n-dimensional real or complex signal π^β from the (possibly noisy) observation of |Ξ¦π^β|, in which Ξ¦ is a matrix of size m Γ n. We consider a high-dimensional setting where n,m ββ with m/n = πͺ(1), and a large class of (possibly correlated) random matrices Ξ¦ and observation channels. Spectral methods are a powerful tool to obtain approximate observations of the signal π^β 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 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 Ξ¦, in an automated manner (i.e. with no optimization over any hyperparameter or preprocessing function).
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