Approximate Message Passing for Sparse Matrices with Application to the Equilibria of Large Ecological Lotka-Volterra Systems

This paper is divided into two parts. The first part is devoted to the studyof a class of Approximate Message Passing (AMP) algorithms which are widelyused in the fields of statistical physics, machine learning, or communicationtheory. The AMP algorithms studied in this part are those where the measurementmatrix has independent elements, up to the symmetry constraint when this matrixis symmetric, with a variance profile that can be sparse. The AMP problem issolved by adapting the approach of Bayati, Lelarge, and Montanari (2015) tothis matrix model. The Lotka-Volterra (LV) model is the standard model forstudying the dynamical behavior of large dimensional ecological food chains.The second part of this paper is focused on the study of the statisticaldistribution of the globally stable equilibrium vector of a LV system in thesituation where the random symmetric interaction matrix among the livingspecies is sparse, and in the regime of large dimensions. This equilibriumvector is the solution of a Linear Complementarity Problem, which distributionis shown to be characterized through the AMP approach developed in the firstpart. In the large dimensional regime, this distribution is close to a mixtureof a large number of truncated Gaussians.