Optimality of Approximate Message Passing Algorithms for Spiked Matrix Models with Rotationally Invariant Noise

TL;DR

This paper introduces a new class of approximate message-passing algorithms tailored for rank-one signal matrix estimation in rotationally invariant noise, providing a precise characterization of their high-dimensional dynamics and optimal denoising strategies.

Contribution

The authors develop a novel AMP algorithm framework that leverages noise structure and signal priors, deriving optimal denoisers and demonstrating minimal asymptotic error within a broad class of algorithms.

Findings

Algorithm achieves minimal asymptotic estimation error.

Explicit characterization of AMP dynamics in high dimensions.

Optimal denoising strategies derived for best performance.

Abstract

We study the problem of estimating a rank one signal matrix from an observed matrix generated by corrupting the signal with additive rotationally invariant noise. We develop a new class of approximate message-passing algorithms for this problem and provide a simple and concise characterization of their dynamics in the high-dimensional limit. At each iteration, these algorithms exploit prior knowledge about the noise structure by applying a non-linear matrix denoiser to the eigenvalues of the observed matrix and prior information regarding the signal structure by applying a non-linear iterate denoiser to the previous iterates generated by the algorithm. We exploit our result on the dynamics of these algorithms to derive the optimal choices for the matrix and iterate denoisers. We show that the resulting algorithm achieves the smallest possible asymptotic estimation error among a broad class of iterative algorithms under a fixed iteration budget.