Elucidating Flow Matching ODE Dynamics with Respect to Data Geometries and Denoisers

• URL: http://arxiv.org/abs/2412.18730v4
• Date: Tue, 03 Jun 2025 03:15:58 GMT
• Title: Elucidating Flow Matching ODE Dynamics with Respect to Data Geometries and Denoisers
• Authors: Zhengchao Wan, Qingsong Wang, Gal Mishne, Yusu Wang,
• Abstract summary: Flow matching (FM) models extend ODE sampler based diffusion models into a general framework.<n>A rigorous theoretical analysis of FM models is essential for sample quality, stability, and broader applicability.<n>In this paper, we advance the theory of FM models through a comprehensive analysis of sample trajectories.
• Score: 10.947094609205765
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Flow matching (FM) models extend ODE sampler based diffusion models into a general framework, significantly reducing sampling steps through learned vector fields. However, the theoretical understanding of FM models, particularly how their sample trajectories interact with underlying data geometry, remains underexplored. A rigorous theoretical analysis of FM ODE is essential for sample quality, stability, and broader applicability. In this paper, we advance the theory of FM models through a comprehensive analysis of sample trajectories. Central to our theory is the discovery that the denoiser, a key component of FM models, guides ODE dynamics through attracting and absorbing behaviors that adapt to the data geometry. We identify and analyze the three stages of ODE evolution: in the initial and intermediate stages, trajectories move toward the mean and local clusters of the data. At the terminal stage, we rigorously establish the convergence of FM ODE under weak assumptions, addressing scenarios where the data lie on a low-dimensional submanifold-cases that previous results could not handle. Our terminal stage analysis offers insights into the memorization phenomenon and establishes equivariance properties of FM ODEs. These findings bridge critical gaps in understanding flow matching models, with practical implications for optimizing sampling strategies and architectures guided by the intrinsic geometry of data.

Related papers

• Consistent World Models via Foresight Diffusion [56.45012929930605]
  We argue that a key bottleneck in learning consistent diffusion-based world models lies in the suboptimal predictive ability.<n>We propose Foresight Diffusion (ForeDiff), a diffusion-based world modeling framework that enhances consistency by decoupling condition understanding from target denoising.
  arXiv  Detail & Related papers  (2025-05-22T10:01:59Z)
• Accelerated Diffusion Models via Speculative Sampling [89.43940130493233]
  Speculative sampling is a popular technique for accelerating inference in Large Language Models. We extend speculative sampling to diffusion models, which generate samples via continuous, vector-valued Markov chains. We propose various drafting strategies, including a simple and effective approach that does not require training a draft model.
  arXiv  Detail & Related papers  (2025-01-09T16:50:16Z)
• A precise asymptotic analysis of learning diffusion models: theory and insights [37.30894159200853]
  We consider the problem of learning a flow or diffusion-based generative model parametrized by a two-layer auto-encoder. We derive a tight characterization of low-dimensional projections of the distribution of samples generated by the learned model.
  arXiv  Detail & Related papers  (2025-01-07T16:56:40Z)
• Provable Statistical Rates for Consistency Diffusion Models [87.28777947976573]
  Despite the state-of-the-art performance, diffusion models are known for their slow sample generation due to the extensive number of steps involved. This paper contributes towards the first statistical theory for consistency models, formulating their training as a distribution discrepancy minimization problem.
  arXiv  Detail & Related papers  (2024-06-23T20:34:18Z)
• Latent diffusion models for parameterization and data assimilation of facies-based geomodels [0.0]
  Diffusion models are trained to generate new geological realizations from input fields characterized by random noise. Latent diffusion models are shown to provide realizations that are visually consistent with samples from geomodeling software.
  arXiv  Detail & Related papers  (2024-06-21T01:32:03Z)
• Flow matching achieves almost minimax optimal convergence [50.38891696297888]
  Flow matching (FM) has gained significant attention as a simulation-free generative model. This paper discusses the convergence properties of FM for large sample size under the $p$-Wasserstein distance. We establish that FM can achieve an almost minimax optimal convergence rate for $1 leq p leq 2$, presenting the first theoretical evidence that FM can reach convergence rates comparable to those of diffusion models.
  arXiv  Detail & Related papers  (2024-05-31T14:54:51Z)
• Categorical Flow Matching on Statistical Manifolds [12.646272756981672]
  We introduce a flow-matching framework on the manifold of parameterized probability measures inspired by information geometry.<n>We develop an efficient training and sampling algorithm that overcomes numerical stability with a diffeomorphism between manifold.<n>We manifest that SFM can learn more complex patterns on the statistical manifold where existing models often fail due to strong prior assumptions.
  arXiv  Detail & Related papers  (2024-05-26T05:50:39Z)
• 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)
• Towards Theoretical Understandings of Self-Consuming Generative Models [56.84592466204185]
  This paper tackles the emerging challenge of training generative models within a self-consuming loop. We construct a theoretical framework to rigorously evaluate how this training procedure impacts the data distributions learned by future models. We present results for kernel density estimation, delivering nuanced insights such as the impact of mixed data training on error propagation.
  arXiv  Detail & Related papers  (2024-02-19T02:08:09Z)
• Explicit Flow Matching: On The Theory of Flow Matching Algorithms with Applications [3.5409403011214295]
  This paper proposes a novel method, Explicit Flow Matching (ExFM), for training and analyzing flow-based generative models. ExFM leverages a theoretically grounded loss function, ExFM loss, to demonstrably reduce variance during training, leading to faster convergence and more stable learning.
  arXiv  Detail & Related papers  (2024-02-05T17:45:12Z)
• A Geometric Perspective on Diffusion Models [57.27857591493788]
  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.
  arXiv  Detail & Related papers  (2023-05-31T15:33:16Z)
• 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)
• Information-Theoretic Diffusion [18.356162596599436]
  Denoising diffusion models have spurred significant gains in density modeling and image generation. We introduce a new mathematical foundation for diffusion models inspired by classic results in information theory.
  arXiv  Detail & Related papers  (2023-02-07T23:03:07Z)
• Moser Flow: Divergence-based Generative Modeling on Manifolds [49.04974733536027]
  Moser Flow (MF) is a new class of generative models within the family of continuous normalizing flows (CNF) MF does not require invoking or backpropagating through an ODE solver during training. We demonstrate for the first time the use of flow models for sampling from general curved surfaces.
  arXiv  Detail & Related papers  (2021-08-18T09:00:24Z)

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.