Universality in polytope phase transitions and message passing algorithms

TL;DR

This paper demonstrates that the high-dimensional behavior of certain nonlinear mappings, including message passing algorithms, is universal and depends only on the first two moments of the random matrix entries, with applications to phase transitions in geometry and compressed sensing.

Contribution

It proves the universality of the behavior of polynomial message passing algorithms across different random matrix ensembles, confirming a conjecture in compressed sensing.

Findings

Universality depends only on first two moments of matrix entries.

Proves phase transition universality in polytope geometry.

Confirms conjecture by Donoho and Tanner for broad random projections.

Abstract

We consider a class of nonlinear mappings $\mathsf{F}_{A,N}$ in $\mathbb{R}^N$ indexed by symmetric random matrices $A\in\mathbb{R}^{N\times N}$ with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Bolthausen [Comm. Math. Phys. 325 (2014) 333-366]. Within information theory, they are known as "approximate message passing" algorithms. We study the high-dimensional (large $N$) behavior of the iterates of $\mathsf{F}$ for polynomial functions $\mathsf{F}$, and prove that it is universal; that is, it depends only on the first two moments of the entries of $A$, under a sub-Gaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves, for a broad class of random projections, a conjecture by David Donoho and Jared Tanner.