@conference {671456,
	title = {Construction of optimal spectral methods in phase retrieval},
	booktitle = {Mathematical and Scientific Machine Learning},
	year = {2021},
	abstract = {We consider the phase retrieval problem, in which the observer wishes to recover a\&nbsp;n-dimensional real or complex signal\&nbsp;X*\&nbsp;from the (possibly noisy) observation of\&nbsp;|ΦX*|, in which\&nbsp;Φ\&nbsp;is a matrix of size\&nbsp;m{\texttimes}n. We consider a \emph{high-dimensional} setting where\&nbsp;n,m{\textrightarrow}$\infty$\&nbsp;with\&nbsp;m/n=O(1), and a large class of (possibly correlated) random matrices\&nbsp;Φ\&nbsp;and observation channels. Spectral methods are a powerful tool to obtain approximate observations of the signal\&nbsp;X*\&nbsp;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\&nbsp;Φ, in an automated manner (i.e. with no optimization over any hyperparameter or preprocessing function).},
	url = {https://arxiv.org/abs/2012.04524},
	author = {Antoine Maillard and Florent Krzakala and Yue M. Lu and Lenka Zdeborova}
}
