Construction of optimal spectral methods in phase retrieval

We consider the phase retrieval problem, in which the observer wishes to recover a n-dimensional real or complex signal X-star from the (possibly noisy) observation of vertical bar Phi X-star vertical bar, in which Phi is a matrix of size m x n. We consider a high-dimensional setting where n, m -> infinity with m/n = O(1), and a large class of (possibly correlated) random matrices Phi and observation channels. Spectral methods are a powerful tool to obtain approximate observations of the signal X-star 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 Phi, in an automated manner (i.e. with no optimization over any hyperparameter or preprocessing function).