Higher-order interactions in statistical physics and machine learning: A model-independent solution to the inverse problem at equilibrium

TL;DR

This paper introduces a universal, model-independent estimator for all-order interactions in equilibrium systems, overcoming limitations of previous methods and applicable across various fields including physics, machine learning, and biology.

Contribution

It presents a non-parametric, unbiased approach using Targeted Learning to infer higher-order interactions without assumptions or approximations, applicable to diverse complex systems.

Findings

Successfully applied to the 2D Ising model and higher-order models

Demonstrated effectiveness on neural network and biological data

Enhanced accuracy by utilizing all available data for interaction estimation

Abstract

The problem of inferring pair-wise and higher-order interactions in complex systems involving large numbers of interacting variables, from observational data, is fundamental to many fields. Known to the statistical physics community as the inverse problem, it has become accessible in recent years due to real and simulated 'big' data being generated. Current approaches to the inverse problem rely on parametric assumptions, physical approximations, e.g. mean-field theory, and ignoring higher-order interactions which may lead to biased or incorrect estimates. We bypass these shortcomings using a cross-disciplinary approach and demonstrate that none of these assumptions and approximations are necessary: We introduce a universal, model-independent, and fundamentally unbiased estimator of all-order symmetric interactions, via the non-parametric framework of Targeted Learning, a subfield of mathematical statistics. Due to its universality, our definition is readily applicable to any system at equilibrium with binary and categorical variables, be it magnetic spins, nodes in a neural network, or protein networks in biology. Our approach is targeted, not requiring fitting unnecessary parameters. Instead, it expends all data on estimating interactions, hence substantially increasing accuracy. We demonstrate the generality of our technique both analytically and numerically on (i) the 2-dimensional Ising model, (ii) an Ising-like model with 4-point interactions, (iii) the Restricted Boltzmann Machine, and (iv) simulated individual-level human DNA variants and representative traits. The latter demonstrates the applicability of this approach to discover epistatic interactions causal of disease in population biomedicine.