Universality of AMP via Tree Pairings

TL;DR

This paper establishes the universality of Approximate Message Passing (AMP) algorithms with polynomial nonlinearities for symmetric sub-Gaussian matrices, using a combinatorial tree pairing approach that extends beyond Gaussian cases.

Contribution

It introduces a combinatorial framework based on tree pairings and Wick algebra to prove universality of AMP for non-Gaussian matrices with polynomial nonlinearities.

Findings

Moments of AMP iterates match state evolution predictions for up to log N iterations.

The combinatorial approach controls contributions from non-Wick-paired trees.

Framework potentially extends to spiked and tensor AMP models.

Abstract

We prove universality for Approximate Message Passing (AMP) with polynomial nonlinearities applied to symmetric sub-Gaussian matrices $A\in\mathbb R^{N\times N}$. Our approach is combinatorial: we represent AMP iterates as sums over trees and define a Wick pairing algebra that counts the number of valid row-wise pairings of edges. The number of such pairings coincides with the trees contribution to the state evolution formulas. This algebra works for non-Gaussian entries. For polynomial nonlinearities of degree at most $D$, we show that the moments of AMP iterates match their state evolution predictions for $t \lesssim \frac{\log N}{D\log D}$ iterations. The proof controls all "excess" trees via explicit enumeration bounds, showing non "Wick-paired" contributions vanish in the large-$N$ limit. The same framework should apply, with some modifications, to spiked AMP and tensor AMP models.