Elliptic Approximate Message Passing and an application to theoretical ecology

Approximate Message Passing (AMP) algorithmshave recently gathered significant attention across disciplines such as statistical physics, machine learning, and communication systems. This study aims to extend AMP algorithms to non-symmetric (elliptic) matrices, motivated by analyzing equilibrium properties in ecological systems featuring elliptic interaction matrices.In this article, we provide the general form of an AMP algorithm associated to a random elliptic matrix, the main change lying in a modification of the corrective (Onsager) term. In order to establish the statistical properties of this algorithm, we use and prove a generalized form of Bolthausen conditioning argument, pivotal to proceed by a Gaussian-based induction.We finally address the initial motivating question from theoretical ecology. Large foodwebs are often described by Lotka-Volterra systems of coupled differential equations, where the interaction matrix is elliptic random. In this context, we design an AMP algorithm to analyze the statistical properties of the equilibrium point in a high-dimensional regime. We rigorously recover the results established by [Bunin, 2017] and [Galla,2018] who used techniques from theoretical physics, and extend them with the help of propagation of chaos type arguments.