A Geometric Perspective on Diffusion Models

• URL: http://arxiv.org/abs/2305.19947v3
• Date: Thu, 22 Aug 2024 15:50:09 GMT
• Title: A Geometric Perspective on Diffusion Models
• Authors: Defang Chen, Zhenyu Zhou, Jian-Ping Mei, Chunhua Shen, Chun Chen, Can Wang,
• Abstract summary: We inspect the ODE-based sampling of a popular variance-exploding SDE. We establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm.
• Score: 57.27857591493788
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Recent years have witnessed significant progress in developing effective training and fast sampling techniques for diffusion models. A remarkable advancement is the use of stochastic differential equations (SDEs) and their marginal-preserving ordinary differential equations (ODEs) to describe data perturbation and generative modeling in a unified framework. In this paper, we carefully inspect the ODE-based sampling of a popular variance-exploding SDE and reveal several intriguing structures of its sampling dynamics. We discover that the data distribution and the noise distribution are smoothly connected with a quasi-linear sampling trajectory and another implicit denoising trajectory that even converges faster. Meanwhile, the denoising trajectory governs the curvature of the corresponding sampling trajectory and its finite differences yield various second-order samplers used in practice. Furthermore, we establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm, with which we can characterize the asymptotic behavior of diffusion models and identify the empirical score deviation. Code is available at \url{https://github.com/zju-pi/diff-sampler}.

Related papers

• Discrete vs. Continuous Trade-offs for Generative Models [0.0]
  This work explores the theoretical and practical foundations of denoising diffusion probabilistic models (DDPMs) DDPMs and score-based generative models, which leverage processes and Brownian motion to model complex data distributions.
  arXiv  Detail & Related papers  (2024-12-26T08:14:27Z)
• Elucidating Flow Matching ODE Dynamics with Respect to Data Geometries [10.947094609205765]
  Diffusion-based generative models have become the standard for image generation. ODE-based samplers and flow matching models improve efficiency, in comparison to diffusion models, by reducing sampling steps through learned vector fields. We advance the theory of flow matching models through a comprehensive analysis of sample trajectories, centered on the denoiser that drives ODE dynamics. Our analysis reveals how trajectories evolve from capturing global data features to local structures, providing the geometric characterization of per-sample behavior in flow matching models.
  arXiv  Detail & Related papers  (2024-12-25T01:17:15Z)
• Diffusing Differentiable Representations [60.72992910766525]
  We introduce a novel, training-free method for sampling differentiable representations (diffreps) using pretrained diffusion models. We identify an implicit constraint on the samples induced by the diffrep and demonstrate that addressing this constraint significantly improves the consistency and detail of the generated objects.
  arXiv  Detail & Related papers  (2024-12-09T20:42:58Z)
• On the Trajectory Regularity of ODE-based Diffusion Sampling [79.17334230868693]
  Diffusion-based generative models use differential equations to establish a smooth connection between a complex data distribution and a tractable prior distribution. In this paper, we identify several intriguing trajectory properties in the ODE-based sampling process of diffusion models.
  arXiv  Detail & Related papers  (2024-05-18T15:59:41Z)
• Gaussian Mixture Solvers for Diffusion Models [84.83349474361204]
  We introduce a novel class of SDE-based solvers called GMS for diffusion models. Our solver outperforms numerous SDE-based solvers in terms of sample quality in image generation and stroke-based synthesis.
  arXiv  Detail & Related papers  (2023-11-02T02:05:38Z)
• Reflected Diffusion Models [93.26107023470979]
  We present Reflected Diffusion Models, which reverse a reflected differential equation evolving on the support of the data. Our approach learns the score function through a generalized score matching loss and extends key components of standard diffusion models.
  arXiv  Detail & Related papers  (2023-04-10T17:54:38Z)
• Score-based Continuous-time Discrete Diffusion Models [102.65769839899315]
  We extend diffusion models to discrete variables by introducing a Markov jump process where the reverse process denoises via a continuous-time Markov chain. We show that an unbiased estimator can be obtained via simple matching the conditional marginal distributions. We demonstrate the effectiveness of the proposed method on a set of synthetic and real-world music and image benchmarks.
  arXiv  Detail & Related papers  (2022-11-30T05:33:29Z)
• An optimal control perspective on diffusion-based generative modeling [9.806130366152194]
  We establish a connection between optimal control and generative models based on differential equations (SDEs) In particular, we derive a Hamilton-Jacobi-Bellman equation that governs the evolution of the log-densities of the underlying SDE marginals. We develop a novel diffusion-based method for sampling from unnormalized densities.
  arXiv  Detail & Related papers  (2022-11-02T17:59:09Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.