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 works 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 performed several controlled numerical experiments where we trained the RBMs using 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.

翻译：生成模型为复杂数据建模提供了直接途径。其中，基于能量的模型通过神经网络架构，旨在精确复现数据中所有统计相关性（以模型玻尔兹曼权重为基准）。然而，理解此类模型的物理内涵仍具挑战性。本研究提出了一种简洁方案，通过构建受限玻尔兹曼机能量函数与包含高阶自旋相互作用的有效伊辛自旋哈密顿量之间的直接映射。该映射涵盖了所有可能阶数的相互作用，超越了传统逆伊辛方法中通常考虑的成对相互作用，从而能够描述复杂数据集。早期研究虽试图实现该目标，但所提出的映射方法未能恰当处理问题复杂性，或缺乏可直接应用于实践的指导方案。为验证方法有效性，我们开展了多项受控数值实验：利用包含局部外场、两体与三体相互作用的预定义模型（处于多种低维拓扑结构中）的平衡样本训练RBM。结果表明，本方法在学习正确相互作用网络方面表现优异，为建模具有研究价值的数据集奠定了应用基础。此外，我们还基于不同训练方法对推断模型的质量进行了评估。