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A Variational Perspective on Diffusion-Based Generative Models and Score Matching (2106.02808v2)

Published 5 Jun 2021 in cs.LG

Abstract: Discrete-time diffusion-based generative models and score matching methods have shown promising results in modeling high-dimensional image data. Recently, Song et al. (2021) show that diffusion processes that transform data into noise can be reversed via learning the score function, i.e. the gradient of the log-density of the perturbed data. They propose to plug the learned score function into an inverse formula to define a generative diffusion process. Despite the empirical success, a theoretical underpinning of this procedure is still lacking. In this work, we approach the (continuous-time) generative diffusion directly and derive a variational framework for likelihood estimation, which includes continuous-time normalizing flows as a special case, and can be seen as an infinitely deep variational autoencoder. Under this framework, we show that minimizing the score-matching loss is equivalent to maximizing a lower bound of the likelihood of the plug-in reverse SDE proposed by Song et al. (2021), bridging the theoretical gap.

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Authors (3)
  1. Chin-Wei Huang (24 papers)
  2. Jae Hyun Lim (5 papers)
  3. Aaron Courville (201 papers)
Citations (172)
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Summary

A Variational Perspective on Diffusion-Based Generative Models and Score Matching

The paper "A Variational Perspective on Diffusion-Based Generative Models and Score Matching" offers an in-depth analysis of diffusion-based generative models through a variational framework, addressing a theoretical gap in score matching and diffusion processes. The paper primarily focuses on high-dimensional image data modeling, where discrete-time diffusion mechanisms have demonstrated empirical success.

Summary of the Paper

The paper centers on the reverse transformation of data to noise using score functions, as highlighted by previous research. It poses significant improvements in understanding these diffusion models by deriving a continuous-time variational framework for likelihood estimation. This generative diffusion process can be interpreted as an infinitely layered variational autoencoder (VAE). Within this framework, minimizing the score-matching loss leads to maximizing a likelihood lower bound, as proposed by Song et al., thus building a theoretical foundation for empirical observations.

Key Contributions

  • Variational Framework: The authors introduce a variational framework designed for continuous-time generative processes. The framework encompasses continuous-time normalizing flows and specific deep VAE cases.
  • Theoretical Insights: By bridging the gap between score matching and likelihood maximization, the paper offers a robust theoretical backing to commonly employed empirical methods.
  • Equivalent Processes: It showcases how alternative processes produce equivalent marginal distributions, contributing to the understanding of stochastic differential equations (SDEs) in generative modeling.
  • Functional Evidence Lower Bound: The derivation of a functional evidence lower bound allows infinite-layer diffusion models, integrating theoretical concepts from stochastic calculus.

Implications and Future Directions

From a theoretical standpoint, this research provides a methodological basis to legitimize the use of diffusion-based models when applying score matching. The implications indicate that generative processes can benefit from score function learning to align closely with observed data distributions.

Practically, the framework offers a path to improve image synthesis models. It suggests that further research could explore optimizing the score-matching approach and investigating plug-in reverse SDEs for various data types. Future developments in AI may see enhanced generative model architectures that leverage this variational perspective, potentially leading to more sophisticated models capable of high-quality image and data generation.

Overall, the paper strengthens the device of score-based generative modeling through a solid theoretical underpinning, advocating for a closer integration of stochastic calculus approaches in advanced machine learning models.