However, training EBMs on data in discrete or mixed state spaces poses significant challenges due to the lack of robust and fast sampling methods. In this work, we propose to train discrete EBMs with Energy Discrepancy, a loss function which only requires the evaluation of the energy function at data points and their perturbed counterparts, thus eliminating the need for Markov chain Monte Carlo. We introduce perturbations of the data distribution by simulating a diffusion process on the discrete state space endowed with a graph structure. This allows us to inform the choice of perturbation from the structure of the modelled discrete variable, while the continuous time parameter enables fine-grained control of the perturbation. Empirically, we demonstrate the efficacy of the proposed approaches in a wide range of applications, including the estimation of discrete densities with non-binary vocabulary and binary image modelling.

Our goal is to develop general-purpose techniques for probabilistic reasoning and learning in structured spaces. These spaces are characterized by complex logical constraints on what constitutes a possible world. We propose a tractable formalism, called probabilistic sentential decision diagrams, and show it effectively learns structured probability distributions in two applications: product configuration and preference learning.