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Learning Energy-Based Generative Models via Potential Flow: A Variational Principle Approach to Probability Density Homotopy Matching

Junn Yong Loo, Michelle Adeline, Julia Kaiwen Lau, Fang Yu Leong, Hwa Hui Tew, Arghya Pal, Vishnu Monn Baskaran, Chee-Ming Ting, Raphaël C. -W. Phan

TL;DR

This work presents Variational Potential Flow Bayes (VPFB), a principled framework for learning energy-based generative models without implicit MCMC or auxiliary networks. It constructs a data-recovery density homotopy and learns a potential flow that matches this marginal with a density-weighted Poisson equation solved via a Deep Ritz variational loss, tying the learned energy to a Boltzmann equilibrium. The approach yields competitive unconditional image generation, robust OOD detection, and coherent interpolation/combination of attributes, while offering theoretical connections to diffusion, flow matching, and Boltzmann EBMs. By enforcing a stationary Boltzmann distribution and leveraging a gradient-based conservative vector field, VPFB enhances interpretability and sampling efficiency, with practical implications for scalable and reliable generative modeling. Future directions include extending to higher-resolution data, exploring long-run MCMC hybrids, and integrating conditional/multi-modal priors to broaden applicability.

Abstract

Energy-based models (EBMs) are a powerful class of probabilistic generative models due to their flexibility and interpretability. However, relationships between potential flows and explicit EBMs remain underexplored, while contrastive divergence training via implicit Markov chain Monte Carlo (MCMC) sampling is often unstable and expensive in high-dimensional settings. In this paper, we propose Variational Potential Flow Bayes (VPFB), a new energy-based generative framework that eliminates the need for implicit MCMC sampling and does not rely on auxiliary networks or cooperative training. VPFB learns an energy-parameterized potential flow by constructing a flow-driven density homotopy that is matched to the data distribution through a variational loss minimizing the Kullback-Leibler divergence between the flow-driven and marginal homotopies. This principled formulation enables robust and efficient generative modeling while preserving the interpretability of EBMs. Experimental results on image generation, interpolation, out-of-distribution detection, and compositional generation confirm the effectiveness of VPFB, showing that our method performs competitively with existing approaches in terms of sample quality and versatility across diverse generative modeling tasks.

Learning Energy-Based Generative Models via Potential Flow: A Variational Principle Approach to Probability Density Homotopy Matching

TL;DR

This work presents Variational Potential Flow Bayes (VPFB), a principled framework for learning energy-based generative models without implicit MCMC or auxiliary networks. It constructs a data-recovery density homotopy and learns a potential flow that matches this marginal with a density-weighted Poisson equation solved via a Deep Ritz variational loss, tying the learned energy to a Boltzmann equilibrium. The approach yields competitive unconditional image generation, robust OOD detection, and coherent interpolation/combination of attributes, while offering theoretical connections to diffusion, flow matching, and Boltzmann EBMs. By enforcing a stationary Boltzmann distribution and leveraging a gradient-based conservative vector field, VPFB enhances interpretability and sampling efficiency, with practical implications for scalable and reliable generative modeling. Future directions include extending to higher-resolution data, exploring long-run MCMC hybrids, and integrating conditional/multi-modal priors to broaden applicability.

Abstract

Energy-based models (EBMs) are a powerful class of probabilistic generative models due to their flexibility and interpretability. However, relationships between potential flows and explicit EBMs remain underexplored, while contrastive divergence training via implicit Markov chain Monte Carlo (MCMC) sampling is often unstable and expensive in high-dimensional settings. In this paper, we propose Variational Potential Flow Bayes (VPFB), a new energy-based generative framework that eliminates the need for implicit MCMC sampling and does not rely on auxiliary networks or cooperative training. VPFB learns an energy-parameterized potential flow by constructing a flow-driven density homotopy that is matched to the data distribution through a variational loss minimizing the Kullback-Leibler divergence between the flow-driven and marginal homotopies. This principled formulation enables robust and efficient generative modeling while preserving the interpretability of EBMs. Experimental results on image generation, interpolation, out-of-distribution detection, and compositional generation confirm the effectiveness of VPFB, showing that our method performs competitively with existing approaches in terms of sample quality and versatility across diverse generative modeling tasks.

Paper Structure

This paper contains 39 sections, 6 theorems, 87 equations, 22 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Related Works
  3. Energy-based Models (EBMs)
  4. Particle Flow
  5. Deep Ritz Method
  6. Variational Potential Flow Bayes (VPFB)
  7. Interpolating Between Prior and Data Likelihood: Log-Homotopy Bayesian Transport
  8. Modeling Potential Flow in a Data-Recovery Homotopy Landscape
  9. Connections to Diffusion Process and Energy-Based Modeling
  10. Variational Potential Energy Loss Formulation: Deep Ritz Method
  11. Training Implementation
  12. Experiments
  13. Density Estimation on 2D Data
  14. Unconditional Image Generation
  15. Image Interpolation and Compositional Generation
  16. ...and 24 more sections

Key Result

Proposition 1

Consider a Gaussian prior $q(x) = \mathcal{N}(x; 0, \omega^2 I)$ and a conditional data likelihood $p(\bar{x} \mid x) = \mathcal{N}(\bar{x}; x, \nu^2 I)$. The log-homotopy transport (eq:unmarginalized_homotopy_h) corresponds to a Gaussian perturbation kernel ${\rho}(x \mid \bar{x},t) = \mathcal{N}(x where $\mathop{\mathrm{\mathrm{sigmoid}}}\nolimits(z) = \frac{1}{1+e^{-z}}$ denotes the logistic (s

Figures (22)

  • Figure 1: 2D potential flow. Top: Sample trajectories from the Gaussian prior noise distribution (black) to the target 2-Moons distribution (blue), driven by the potential energy $\Phi(x,t)$ and sampled using an ODE solver. Bottom: Time evolution of the learned potential energy landscape $\Phi(x,t)$.
  • Figure 2: 2D Boltzmann density estimation. Top: Sample trajectories from the Gaussian prior noise distribution (black) to the target 2-Moons distribution (blue), driven by the Boltzmann energy and sampled via SGLD. Middle: Visualization of the log-density estimation (up to an additive constant) $\log p_B (x) = {\Phi}_{B}(x)$ parameterized by Boltzmann energy.
  • Figure 3: Uncurated and unconditional samples generated for CIFAR-10 (left) and CelebA (right).
  • Figure 4: Compositional and conditional CelebA samples generated based on three attribute pairs.
  • Figure 5: Generated CIFAR-10 samples and their five nearest neighbors in train set based on pixel distance.
  • ...and 17 more figures

Theorems & Definitions (18)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • ...and 8 more