Abstract

In this talk, I will present our recent work on the optimality of Approximate Message Passing (AMP) algorithms for spiked matrix models with rotationally invariant noise. We introduce a new AMP algorithm that employs a matrix denoiser—acting on the eigenvalues of the observed matrix—and an iterate denoiser—applied to the AMP iterates. The resulting dynamics admit a simple state-evolution characterization, which allows us to identify the optimal pair of denoisers achieving the minimum possible asymptotic estimation error among a broad class of iterative algorithms. I will also discuss ongoing work that extends this framework to rectangular spiked matrix models, where we develop an AMP algorithm with optimal spectral initialization, further broadening the scope and applicability of the theory.