Unifying AMP Algorithms for Rotationally-Invariant Models

TL;DR

This paper develops a unified framework for AMP algorithms in rotationally-invariant models, deriving correct Onsager terms and introducing new variants, with applications to spiked model estimation.

Contribution

It provides a systematic derivation of AMP algorithms using a general iterative template and introduces novel AMP variants based on free cumulants.

Findings

Successfully rederived existing AMP algorithms with a unified approach

Introduced two new AMP variants demonstrating framework flexibility

Applied new algorithms to estimation in spiked models

Abstract

This paper presents a unified framework for constructing Approximate Message Passing (AMP) algorithms for rotationally-invariant models. By employing a general iterative algorithm template and reducing it to long-memory Orthogonal AMP (OAMP), we systematically derive the correct Onsager terms of AMP algorithms. This approach allows us to rederive an AMP algorithm introduced by Fan and Opper et al., while shedding new light on the role of free cumulants of the spectral law. The free cumulants arise naturally from a recursive centering operation, potentially of independent interest beyond the scope of AMP. To illustrate the flexibility of our framework, we introduce two novel AMP variants and apply them to estimation in spiked models.