Approximate Message Passing for general non-Symmetric random matrices

TL;DR

This work extends Approximate Message Passing (AMP) to general non-symmetric random matrices with variance and correlation profiles, enabling analysis of high-dimensional systems beyond symmetric or elliptic models. By formulating a density evolution (DE) framework that tracks the joint statistics of AMP iterates via covariance objects $\{R^t\}$, the authors show convergence of empirical measures to Gaussian limits described by DE, with the Onsager correction carefully chosen to account for the non-symmetric, possibly sparse structure. They develop a robust proof strategy based on combinatorial non-backtracking trees for polynomial activations, then extend to general activations through density arguments and polynomial approximations, and finally handle non-centered and nonzero-diagonal models through perturbation arguments. The results provide a rigorous, general toolkit for predicting AMP dynamics in complex ecosystems (e.g., Lotka–Volterra models) and other applications where non-symmetric interactions with variance/correlation structure arise, offering precise asymptotics for the distribution of iterates and enabling principled performance analysis. Overall, this work broadens the scope of AMP, delivering a principled framework to analyze high-dimensional estimation and dynamical systems with rich randomness structures.

Abstract

Approximate Message Passing (AMP) algorithms are a family of iterative algorithms based on large random matrices with the special property of tracking the statistical properties of their iterates. They are used in various fields such as Statistical Physics, Machine learning, Communication systems, Theoretical ecology, etc. In this article we consider AMP algorithms based on non-Symmetric random matrices with a general variance profile, possibly sparse, a general covariance profile, and non-Gaussian entries. We hence substantially extend the results on Elliptic random matrices that we developed in [Gueddari et al., 2024]. From a technical point of view, we enhance the combinatorial techniques developed in [Bayati et al., 2015] and in [Hachem, 2024]. Our main motivation is the understanding of equilibria of large food-webs described by Lotka-Volterra systems of ODE, in the continuation of the works of [Hachem, 2024], [Akjouj et al., 2024] and [Gueddari et al., 2024], but the versatility of the model studied might be of interest beyond these particular applications.

Figures (4)

• Figure 1: The Gaussian matrix $(\boldsymbol{Z}^1,\cdots, \boldsymbol{Z}^t)$, the notations $\boldsymbol{Z}^t$ and $\vec{Z}^t_i$. Rows $Z_i=(Z_i^t,\, t\ge 1)$ are independent. The correlations within each row are described by the DE equations: $\vec{Z}_i^t\sim {\mathcal{N}}_t(0,R_i^t)$, see Definition \ref{['def:de']}.
• Figure 2: Proof steps.
• Figure 3: Example of a tree $T\in\overline{\mathcal{T}}^3$ for parameters $q=2$, $d=4$, $t=3$ and $n=6$. The types are written between braces, the marks are between brackets and leafs info is between parentheses. In this example, $T$ is not a non-backtracking tree because of the two paths $(a\leftarrow b \leftarrow c)$ and $(b\leftarrow d \leftarrow e)$.
• Figure 4: Tree structure.

Theorems & Definitions (57)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Remark 1.4
• Definition 2.1
• Remark 2.2
• Remark 2.3: versatility
• Remark 2.4: on Assumption A-\ref{['ass:nu']}
• Theorem 2.1
• Remark 2.5