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 $|mathbf{Phi} mathbf{X}^star|$, 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