Flow matching achieves minimax optimal convergence
- URL: http://arxiv.org/abs/2405.20879v1
- Date: Fri, 31 May 2024 14:54:51 GMT
- Title: Flow matching achieves minimax optimal convergence
- Authors: Kenji Fukumizu, Taiji Suzuki, Noboru Isobe, Kazusato Oko, Masanori Koyama,
- Abstract summary: Flow matching (FM) has gained significant attention as a simulation-free generative model.
This paper discusses the convergence properties of FM in terms of the $p$-Wasserstein distance, a measure of distributional discrepancy.
We establish that FM can achieve the minmax 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.
- Score: 50.38891696297888
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Flow matching (FM) has gained significant attention as a simulation-free generative model. Unlike diffusion models, which are based on stochastic differential equations, FM employs a simpler approach by solving an ordinary differential equation with an initial condition from a normal distribution, thus streamlining the sample generation process. This paper discusses the convergence properties of FM in terms of the $p$-Wasserstein distance, a measure of distributional discrepancy. We establish that FM can achieve the minmax 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. Our analysis extends existing frameworks by examining a broader class of mean and variance functions for the vector fields and identifies specific conditions necessary to attain these optimal rates.
Related papers
- Switched Flow Matching: Eliminating Singularities via Switching ODEs [12.273757042838675]
Continuous-time generative models, such as Flow Matching (FM), construct probability paths to transport between one distribution and another.
During inference, however, the learned model often requires multiple neural network evaluations to accurately integrate the flow.
We propose Switched FM (SFM), that eliminates singularities via switching ODEs, as opposed to using a uniform ODE in FM.
arXiv Detail & Related papers (2024-05-19T16:21:04Z) - 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) - Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative
Models [49.81937966106691]
We develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models.
In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach.
arXiv Detail & Related papers (2023-06-15T16:30:08Z) - A Geometric Perspective on Diffusion Models [60.69328526215776]
We inspect the ODE-based sampling of a popular variance-exploding SDE and reveal several intriguing structures of its sampling dynamics.
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) - Diffusion Models are Minimax Optimal Distribution Estimators [49.47503258639454]
We provide the first rigorous analysis on approximation and generalization abilities of diffusion modeling.
We show that when the true density function belongs to the Besov space and the empirical score matching loss is properly minimized, the generated data distribution achieves the nearly minimax optimal estimation rates.
arXiv Detail & Related papers (2023-03-03T11:31:55Z) - Improving and generalizing flow-based generative models with minibatch
optimal transport [90.01613198337833]
We introduce the generalized conditional flow matching (CFM) technique for continuous normalizing flows (CNFs)
CFM features a stable regression objective like that used to train the flow in diffusion models but enjoys the efficient inference of deterministic flow models.
A variant of our objective is optimal transport CFM (OT-CFM), which creates simpler flows that are more stable to train and lead to faster inference.
arXiv Detail & Related papers (2023-02-01T14:47:17Z) - 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) - Efficient CDF Approximations for Normalizing Flows [64.60846767084877]
We build upon the diffeomorphic properties of normalizing flows to estimate the cumulative distribution function (CDF) over a closed region.
Our experiments on popular flow architectures and UCI datasets show a marked improvement in sample efficiency as compared to traditional estimators.
arXiv Detail & Related papers (2022-02-23T06:11:49Z)
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.