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.

翻译：生成模型为复杂数据的建模提供了直接途径。基于能量的模型试图在神经网络形式的能量函数对应的玻尔兹曼权重层面，对数据中观测到的统计相关性进行编码。本研究旨在应对理解此类模型物理诠释的挑战。我们提出一种简洁方案，通过建立受限玻尔兹曼机与有效伊辛自旋哈密顿量之间的直接映射关系来实现目标。该映射包含所有可能阶数的相互作用，突破了逆伊辛（或玻尔兹曼机）方法中通常考虑的成对相互作用局限，从而能够描述复杂数据集。早期工作虽尝试实现该目标，但所提出的映射在推断应用中不够精确，未能恰当处理问题的复杂性，或未提供适用于实际应用的精准操作规范。为验证方法有效性，我们开展了多次受控逆数值实验：利用具有局域外场、不同稀疏拓扑结构下二体与三体相互作用的预定义模型平衡样本对RBM进行训练。结果表明，我们提出的方法在学习正确相互作用网络方面具有有效性，为将其应用于有趣的二值变量数据集建模铺平了道路。我们还基于不同训练方法对推断模型的质量进行了评估。