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 ${\mathbf{X}}^{\star }$ from the (possibly noisy) observation of $\mathrm{\mid }\mathbf{\Phi }{\mathbf{X}}^{\star }\mathrm{\mid }$, in which $\mathbf{\Phi }$ is a matrix of size $m\times n$. We consider a \emph{high-dimensional} setting where $n,m\to \mathrm{\infty }$ with $m/n = $\mathcal{O}$(1)$,andalargeclassof(possiblycorrelated)randommatrices$$\mathbf{\Phi }$$ 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).