Inferring Higher-Order Couplings with Neural Networks

Maximum entropy methods, rooted in the inverse Ising or Potts problem of statistical physics, have been fundamental in inferring pairwise interaction models of complex systems. These models have found impactful applications in fields such as bioinformatics, where they are used to infer protein structural contacts, and in neuroscience, for describing patterns of neural activity. Despite their considerable success, these approaches fail to capture higher-order interactions that could be crucial for a full description of collective behavior. In contrast, modern machine learning methods can model such interactions, but their interpretability usually comes at a prohibitive computational cost. Restricted Boltzmann machines (RBMs) provide a computationally efficient alternative by encoding statistical correlations through hidden units in a bipartite architecture. In this Letter, we introduce a method that maps RBMs onto generalized Potts models, enabling the systematic extraction of interactions up to arbitrary order. Leveraging large-N approximations, made tractable by the RBM's structure, we extract effective many-body couplings with minimal computational effort. We further propose a robust framework for recovering higher-order interactions in more complex generative models, and introduce a simple gauge-fixing scheme for the effective Potts representation. Validation on synthetic data demonstrates accurate recovery of two- and three-body interactions. Applied to protein sequence data, our method reconstructs contact maps with high fidelity and outperforms state-of-the-art inverse Potts models. These results establish RBMs as a powerful and efficient tool for modeling higher-order structure for high-dimensional categorical data.