Generative models offer a direct way to model complex data. Among them,
energy-based models provide us with a neural network model that aims to
accurately reproduce all statistical correlations observed in the data at the
level of the Boltzmann weight of the model. However, one challenge is to
understand the physical interpretation of such models. In this study, we
propose a simple solution by implementing a direct mapping between the energy
function of the Restricted Boltzmann Machine and an effective Ising spin
Hamiltonian that includes high-order interactions between spins. This mapping
includes interactions of all possible orders, going beyond the conventional
pairwise interactions typically considered in the inverse Ising approach, and
allowing the description of complex datasets. Earlier work attempted to achieve
this goal, but the proposed mappings did not do properly treat the complexity
of the problem or did not contain direct prescriptions for practical
application. To validate our method, we perform several controlled numerical
experiments where the training samples are equilibrium samples of predefined
models containing local external fields, two-body and three-body interactions
in various low-dimensional 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 datasets.
We also evaluate the quality of the inferred model based on different training
methods.Comment: 16 figures, 22 page