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Elliptic Approximate Message Passing and an application to theoretical ecology

Mohammed-Younes Gueddari, Walid Hachem, Jamal Najim

TL;DR

This article provides 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, and rigorously recovers the results established by [Bunin, 2017] and [Galla, 2018] and extend them with the help of propagation of chaos type arguments.

Abstract

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.

Elliptic Approximate Message Passing and an application to theoretical ecology

TL;DR

This article provides 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, and rigorously recovers the results established by [Bunin, 2017] and [Galla, 2018] and extend them with the help of propagation of chaos type arguments.

Abstract

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.
Paper Structure (23 sections, 25 theorems, 237 equations, 2 figures)

This paper contains 23 sections, 25 theorems, 237 equations, 2 figures.

Table of Contents

  1. Introduction
  2. AMP for random elliptic matrices
  3. Background
  4. Assumptions
  5. Main result
  6. Application to theoretical ecology: equilibria of large LV systems
  7. Large Lotka-Volterra systems
  8. Existence of a stable and unique equilibrium
  9. Statistical properties of the LV equilibrium $\boldsymbol{x}^\star$
  10. Propagation of chaos
  11. Proofs of Theorem \ref{['th:main']} and Corollary \ref{['cor:partialAMP']}
  12. Preliminary results on elliptic matrices
  13. Proof of Theorem \ref{['th:main']}: notations and some preparation
  14. Proof of Theorem \ref{['th:main']}: adaptation of Bolthausen conditioning argument
  15. Proof of Theorem \ref{['th:main']}: end of proof
  16. ...and 8 more sections

Key Result

Theorem 1

Let $A_n$ be a $n\times n$ matrix such that $\sqrt{n}A_n$ is drawn from the GOE. Let $\boldsymbol{u}^0, \boldsymbol{b}\in \mathbb{R}^n$ independent from $A_n$, and $(\boldsymbol{u}^\ell)_{1\le \ell\le k}$ be defined by eq:AMP_symmetric. Suppose that $\mu^{\boldsymbol{u}^0,\boldsymbol{b}} \xrightarro

Figures (2)

  • Figure 1: Comparison between the theoretical solution of the fixed point equations \ref{['eq:sys']} and their empirical Monte Carlo counterpart obtained by computing equilibria $\boldsymbol{x}^\star$ for various realizations of matrix $A$. Every $\boldsymbol{x}^\star$ is the solution of a Linear Complementarity Problem (see \ref{['eq:LCP-first-sight']}) and is thus computed by Lemke algorithm. For Figure \ref{['fig:prop']} and Figure \ref{['fig:dist']} we chose a matrix of size $200$ and we fixed the number of Monte Carlo experiments to $100$ and $500$ respectively.
  • Figure 2: The vector $\boldsymbol{r}$ is decomposed into three consecutive blocks, for each block we fix a constant value $r_1,r_2,r_3 = 1,3,6$ respectively, we also choose different sizes of the blocks to be $n/2, 3n/10, n/5$. We then solve the Linear Complementarity Problem problem (see \ref{['eq:LCP-first-sight']}) with the help of Lemke algorithm for $\rho=0$ and $\kappa=2$.

Theorems & Definitions (55)

  • Theorem : bayati2011dynamicsfeng2021unifying
  • Definition 1.1: Gaussian elliptic matrix model
  • Lemma 2.1
  • Remark 2.2
  • Theorem 1
  • Remark 2.3
  • Remark 2.4
  • Corollary 2: blockwise AMP
  • Proposition 3.1: Prop. 2.3 in clenet2022equilibrium
  • Remark 3.2
  • ...and 45 more