Orthogonal Approximate Message Passing with Optimal Spectral Initializations for Rectangular Spiked Matrix Models
This work addresses signal estimation problems in high-dimensional statistics, particularly for rectangular matrices, but appears incremental as it builds on existing OAMP frameworks.
The authors tackled signal estimation in rectangular spiked matrix models with rotationally invariant noise by proposing an orthogonal approximate message passing algorithm with optimal spectral initializations, achieving performance that matches replica-symmetric predictions for the Bayes-optimal estimator.
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.