Energy-Based Modelling for Discrete and Mixed Data via Heat Equations on Structured Spaces (2412.01019v1)

Abstract: Energy-based models (EBMs) offer a flexible framework for probabilistic modelling across various data domains. 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. Finally, we train EBMs on tabular data sets with applications in synthetic data generation and calibrated classification.

Energy-Based Modelling for Discrete and Mixed Data via Heat Equations on Structured Spaces

The paper, "Energy-Based Modelling for Discrete and Mixed Data via Heat Equations on Structured Spaces," introduces an innovative approach to training energy-based models (EBMs) on discrete and mixed data types. The paper carefully constructs a methodology leveraging perturbations defined via heat equations on graph structures to circumvent the prevalent issues of inefficient sampling and intractable normalization inherent in traditional EBM training.

EBMs have garnered significant attention due to their flexibility and potential in generative modeling, particularly in discrete domains. However, these models typically depend on Markov Chain Monte Carlo (MCMC) methods like contrastive divergence, which often face challenges concerning mixing speed and convergence guarantees. This paper proposes an alternative via the Energy Discrepancy (ED) loss, sidestepping gradient and MCMC dependencies by focusing solely on the functional evaluations of the energy landscape. This approach not only eases computational burdens but also complements traditional contrastive divergence by addressing its limitations in sampling.

The authors make significant contributions by presenting a novel perspective on discrete diffusion processes and leveraging graph-based heat equations. By constructing perturbations on structured state spaces, their method enables more controlled, efficient, and accurate training. The paper demonstrates this through rigorous theoretical frameworks and provides empirical evidence showcasing improved performance against baseline methods across diverse applications, including discrete density estimation, synthetic tabular data generation, and calibrated classification tasks.

Key Contributions and Results:

1. The development of discrete diffusion processes via a heat equation on graph structures marks a significant advancement. The authors effectively exploit the underlying graph geometry to refine perturbation techniques, a method that allows for a fine-grained balance between perturbation scale and variance in gradients.
2. The paper addresses the challenges presented by mixed state spaces, introducing the first robust EBM training method applicable to tabular datasets. The empirical results highlight the method's effectiveness in synthetic data generation and classifications, presenting an avenue for more refined generative models.
3. In terms of performance, the energy discrepancy approach provides competitive results against state-of-the-art EBM training methods, offering computational efficiency due to its non-reliance on iterative MCMC-based methods. This is particularly evident in experiments involving discrete image modeling, where the proposed method reduces running time substantially compared to traditional methods.

Practical and Theoretical Implications:

The practical implications of this paper are wide-ranging. For real-world applications, especially those involving synthetic data and generative modeling, this methodology opens avenues for deeper integration of graphical structures in data representation. This approach is particularly beneficial for high-dimensional discrete problems, providing a reliable alternative to conventional sampling-laden methods.

Theoretically, the construction of perturbations via heat equations on structured spaces could inspire future research on modeling complex discrete and continuous combinations, potentially extending into structured data realms like molecular structures and linguistic patterns. By efficiently converging to maximum likelihood estimations, the methodology sets a precedent for future research aimed at reducing computational overhead without sacrificing model accuracy.

In conclusion, the paper presents a substantial advance in EBM training through a novel approach of perturbations informed by discrete diffusion and graph structures. While the results are promising, future research is essential to explore further applications and refine methodologies for more complex and constraints-heavy data domains. This paper serves as a foundation for ongoing exploration into more scalable, reliable, and versatile generative modeling techniques.