Construction of optimal spectral methods in phase retrieval

We consider the \emph{phase retrieval} problem, in which the observer wishes to recover a $n$-dimensional real or complex signal $\bX^\star$ from the (possibly noisy) observation of $|\bPhi \bX^\star|$, in which $\bPhi$ 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 $\bPhi$ and observation channels. Spectral methods are a powerful tool to obtain approximate observations of the signal $\bX^\star$ 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 $\bPhi$, in an automated manner (i.e. with no optimization over any hyperparameter or preprocessing function).