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