Orthogonal Approximate Message Passing with Optimal Spectral Initializations for Rectangular Spiked Matrix Models

TL;DR

This paper introduces an orthogonal approximate message passing algorithm for rectangular spiked matrix models with rotationally invariant noise, providing optimal spectral initializations and rigorous performance analysis.

Contribution

It develops an OAMP algorithm with state evolution for general RI noise, enabling optimal denoising and outlier combination, advancing high-dimensional signal estimation methods.

Findings

Algorithm's performance matches replica-symmetric predictions.

Spectral initializations are effective under minimal noise assumptions.

Proposed method is conjectured to be statistically optimal among iterative estimators.

Abstract

We propose an orthogonal approximate message passing (OAMP) algorithm for signal estimation in the rectangular spiked matrix model with general rotationally invariant (RI) noise. We establish a rigorous state evolution that precisely characterizes the algorithm's high-dimensional dynamics and enables the construction of iteration-wise optimal denoisers. Within this framework, we accommodate spectral initializations under minimal assumptions on the empirical noise spectrum. In the rectangular setting, where a single rank-one component typically generates multiple informative outliers, we further propose a procedure for combining these outliers under mild non-Gaussian signal assumptions. For general RI noise models, the predicted performance of the proposed optimal OAMP algorithm agrees with replica-symmetric predictions for the associated Bayes-optimal estimator, and we conjecture that it is statistically optimal within a broad class of iterative estimation methods.