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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).
We present the optimal design of a spectral method widely used to initialize nonconvex optimization algorithms for solving phase retrieval and other signal recovery problems. Our work leverages recent results that provide an exact characterization of
Recently, it was discovered by several authors that a $q$-ary optimal locally recoverable code, i.e., a locally recoverable code archiving the Singleton-type bound, can have length much bigger than $q+1$. This is quite different from the classical $q
We consider the phase retrieval problem of reconstructing a $n$-dimensional real or complex signal $mathbf{X}^{star}$ from $m$ (possibly noisy) observations $Y_mu = | sum_{i=1}^n Phi_{mu i} X^{star}_i/sqrt{n}|$, for a large class of correlated real a
This paper investigates the convergence of the randomized Kaczmarz algorithm for the problem of phase retrieval of complex-valued objects. While this algorithm has been studied for the real-valued case}, its generalization to the complex-valued case
This paper is concerned with stable phase retrieval for a family of phase retrieval models we name locally stable and conditionally connected (LSCC) measurement schemes. For every signal $f$, we associate a corresponding weighted graph $G_f$, defined