Approximate Message Passing for general non-Symmetric random matrices

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.