Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling (2504.10612v3)

Abstract: The most widely used generative models map noise and data distributions by matching flows or scores. However, they struggle to incorporate partial observations and additional priors--something energy-based models (EBMs) handle elegantly by simply adding corresponding scalar energy terms. We address this issue by proposing Energy Matching, a framework that endows flow-based approaches with the flexibility of EBMs. Far from the data manifold, samples move along curl-free, optimal transport paths from noise to data. As they approach the data manifold, an entropic energy term guides the system into a Boltzmann equilibrium distribution, explicitly capturing the underlying likelihood structure of the data. We parameterize this dynamic with a single time-independent scalar field, which serves as both a powerful generator and a flexible prior for effective regularization of inverse problems. Our method substantially outperforms existing EBMs on CIFAR-10 and ImageNet generation in terms of fidelity, while retaining simulation-free training of transport-based approaches away from the data manifold. Furthermore, we leverage the method's flexibility to introduce an interaction energy that supports diverse mode exploration, which we demonstrate in a controlled protein-generation setting. Our approach focuses on learning a scalar potential energy--without time-conditioning, auxiliary generators, or additional networks--which marks a significant departure from recent EBM methods. We believe that this simplified framework significantly advances EBMs capabilities and paves the way for their wider adoption in generative modeling across diverse domains.

• The paper introduces Energy Matching, a novel framework unifying flow matching and energy-based models via a scalar potential energy field to enhance generative dynamics and likelihood modeling.
• It achieves state-of-the-art generative quality, demonstrating a FID of 3.97 on CIFAR-10 while utilizing a simplified, time-independent scalar field architecture.
• The framework includes an interaction energy term for mode exploration and enables inverse problem solving through explicit likelihood modeling.

Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling

The paper "Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling" proposes a novel framework termed Energy Matching, which aims to enhance generative modeling by integrating flow-based approaches with the expressiveness of energy-based models (EBMs). This research is motivated by the limitations observed in conventional generative models that map noise to data through flow matching or energy-based techniques, particularly in handling partial observations or additional priors.

Key Contributions and Methodology

The Energy Matching framework distinguishes itself by utilizing a scalar potential energy field to parameterize generative dynamics. This innovative approach ensures that samples are guided through optimal transport paths from noise to data manifolds, employing an entropic energy component to achieve Boltzmann equilibrium distributions as they near the data manifold. This distinct separation of flow and energy phases facilitates the creation of a generator that merges the production efficiency typical of flow methods with the robust likelihood modeling inherent to EBMs.

Some salient features of the Energy Matching approach include:

1. Enhanced Performance on CIFAR-10: Demonstrates a substantial improvement with a Fréchet Inception Distance (FID) of 3.97, significantly outperforming traditional EBMs which score 8.61.
2. Interaction Energy for Mode Exploration: The framework introduces an additional interaction energy term that allows for diverse exploration of modes within the data distribution.
3. Simplified Architecture: Utilizing a single, time-independent scalar field breaks from the time-conditioned and often complex architectures of recent EBMs, simplifying the training and application.

Theoretical Foundations

The methodology leverages recent advances in Wasserstein gradient flows, particularly the Jordan–Kinderlehrer–Otto (JKO) scheme. The paper provides a thorough explanation of how the discrete-time evolution of a probability distribution can be efficiently managed within this framework, with the energy component explicitly capturing the likelihood of data.

The framework suggests a training split into two regimes:

• Away from data manifold: The methodology emphasizes a flow-like, deterministic process that transports samples efficiently.
• Near the data manifold: It transitions to a contrastive divergence approach, refining the energy potential to accurately represent the data distribution via a learned scalar field.

Practical and Theoretical Implications

The implications of this work span both practical and theoretical realms:

• Generative Quality: Empirical findings suggest that Energy Matching outcompetes many established approaches, presenting an enticing computational trade-off by minimizing network complexity while enhancing simulation stability.
• Inverse Problem Solving: The framework's explicit likelihood modeling facilitates its application in solving inverse problems, integrating well-defined priors into the modeling process.
• Local Intrinsic Dimension Estimation: By analyzing the Hessian spectrum of the learned energy field, this approach offers insightful metrics about the complexity and dimensional structure of data.

Future Directions

The paper opens several potential avenues for further research and development:

• Extending applications to more complex data modalities and domains where interpretability and control of generative dynamics are crucial.
• Refining computational efficiency, especially concerning Hessian calculations in high-dimensional data tasks.
• Exploring synergies with other generative frameworks, such as adversarial or transformer-based models.

In conclusion, Energy Matching presents a compelling synthesis of flow matching and energy-based models, offering a robust framework that effectively balances simplicity with powerful generative capabilities. Its contributions lay a promising foundation for advancements across various fields requiring high-fidelity data generation and manipulation.