Alfonso de Jesús Navas Gómez

Aurélien Decelle, Alfonso de Jesús Navas Gómez, Beatriz Seoane

Energy-based learning is a powerful framework for generative modelling, but its training is inherently non-convex, leading potentially to sensitivity to initialisation, poor local optima, and unstable gradient dynamics. We present a dynamical analysis of energy-based learning through the lens of the effective model, which can be interpreted as either a generalised Ising model with higher-order interactions or the Fourier expansion of the energy. Under sufficient expressivity, we show that the gradient flow induced by learning strictly positive distributions over binary variables admits two types of fixed points: data-consistent points, which exactly reproduce the target distribution, and spurious points, which satisfy stationarity without matching the target distribution. Around data-consistent points, we show that perturbations are either stable or neutral, with neutral directions leaving the effective model invariant. Finally, we show that gradient dynamics induce a hierarchy in which lower-order interactions are learned before higher-order ones. This provides a mechanistic explanation for the distributional simplicity bias and clarifies why fixed points that are not data-consistent at low orders are not observed in practice.

Aurélien Decelle, Alfonso de Jesús Navas Gómez, Beatriz Seoane

Maximum entropy methods, rooted in the inverse Ising/Potts problem from statistical physics, are widely used to model pairwise interactions in complex systems across disciplines such as bioinformatics and neuroscience. While successful, these approaches often fail to capture higher-order interactions that are critical for understanding collective behavior. In contrast, modern machine learning methods can model such interactions, but their interpretability often 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 work, 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 in high-dimensional categorical data.