Alfonso de Jesús Navas Gómez

Aurélien Decelle, Cyril Furtlehner, Alfonso De Jesus Navas Gómez et al.

Generative models offer a direct way of modeling complex data. Energy-based models attempt to encode the statistical correlations observed in the data at the level of the Boltzmann weight associated with an energy function in the form of a neural network. We address here the challenge of understanding the physical interpretation of such models. In this study, we propose a simple solution by implementing a direct mapping between the Restricted Boltzmann Machine and an effective Ising spin Hamiltonian. This mapping includes interactions of all possible orders, going beyond the conventional pairwise interactions typically considered in the inverse Ising (or Boltzmann Machine) approach, and allowing the description of complex datasets. Earlier works attempted to achieve this goal, but the proposed mappings were inaccurate for inference applications, did not properly treat the complexity of the problem, or did not provide precise prescriptions for practical application. To validate our method, we performed several controlled inverse numerical experiments in which we trained the RBMs using equilibrium samples of predefined models with local external fields, 2-body and 3-body interactions in different sparse topologies. The results demonstrate the effectiveness of our proposed approach in learning the correct interaction network and pave the way for its application in modeling interesting binary variable datasets. We also evaluate the quality of the inferred model based on different training methods.

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.