Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans

• URL: http://arxiv.org/abs/2501.16839v3
• Date: Thu, 13 Feb 2025 13:39:26 GMT
• Title: Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans
• Authors: Christian Wald, Gabriele Steidl,
• Abstract summary: Flow matching techniques can be used to solve inverse problems. We show how flow matching can be used for solving inverse problems. We briefly address continuous normalizing flows and score matching techniques.
• Score: 1.9766522384767222
• License:
• Abstract: Among generative neural models, flow matching techniques stand out for their simple applicability and good scaling properties. Here, velocity fields of curves connecting a simple latent and a target distribution are learned. Then the corresponding ordinary differential equation can be used to sample from a target distribution, starting in samples from the latent one. This paper reviews from a mathematical point of view different techniques to learn the velocity fields of absolutely continuous curves in the Wasserstein geometry. We show how the velocity fields can be characterized and learned via i) transport plans (couplings) between latent and target distributions, ii) Markov kernels and iii) stochastic processes, where the latter two include the coupling approach, but are in general broader. Besides this main goal, we show how flow matching can be used for solving Bayesian inverse problems, where the definition of conditional Wasserstein distances plays a central role. Finally, we briefly address continuous normalizing flows and score matching techniques, which approach the learning of velocity fields of curves from other directions.

Related papers

• Variational Rectified Flow Matching [100.63726791602049]
  Variational Rectified Flow Matching enhances classic rectified flow matching by modeling multi-modal velocity vector-fields. We show on synthetic data that variational rectified flow matching leads to compelling results.
  arXiv  Detail & Related papers  (2025-02-13T18:59:15Z)
• 2-Rectifications are Enough for Straight Flows: A Theoretical Insight into Wasserstein Convergence [54.580605276017096]
  We provide the first theoretical analysis of the Wasserstein distance between the sampling distribution of Rectified Flow and the target distribution. We show that for a rectified flow from a Gaussian to any general target distribution with finite first moment, two rectifications are sufficient to achieve a straight flow.
  arXiv  Detail & Related papers  (2024-10-19T02:36:11Z)
• Consistency Flow Matching: Defining Straight Flows with Velocity Consistency [97.28511135503176]
  We introduce Consistency Flow Matching (Consistency-FM), a novel FM method that explicitly enforces self-consistency in the velocity field. Preliminary experiments demonstrate that our Consistency-FM significantly improves training efficiency by converging 4.4x faster than consistency models.
  arXiv  Detail & Related papers  (2024-07-02T16:15:37Z)
• Flow Map Matching [15.520853806024943]
  Flow map matching is an algorithm that learns the two-time flow map of an underlying ordinary differential equation. We show that flow map matching leads to high-quality samples with significantly reduced sampling cost compared to diffusion or interpolant methods.
  arXiv  Detail & Related papers  (2024-06-11T17:41:26Z)
• Optimal Flow Matching: Learning Straight Trajectories in Just One Step [89.37027530300617]
  We develop and theoretically justify the novel textbf Optimal Flow Matching (OFM) approach. It allows recovering the straight OT displacement for the quadratic transport in just one FM step. The main idea of our approach is the employment of vector field for FM which are parameterized by convex functions.
  arXiv  Detail & Related papers  (2024-03-19T19:44:54Z)
• Sampling via Gradient Flows in the Space of Probability Measures [10.892894776497165]
  Recent work shows that algorithms derived by considering gradient flows in the space of probability measures open up new avenues for algorithm development. This paper makes three contributions to this sampling approach by scrutinizing the design components of such gradient flows.
  arXiv  Detail & Related papers  (2023-10-05T15:20:35Z)
• Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimization [87.21285093582446]
  Diffusion Generative Flow Samplers (DGFS) is a sampling-based framework where the learning process can be tractably broken down into short partial trajectory segments. Our method takes inspiration from the theory developed for generative flow networks (GFlowNets)
  arXiv  Detail & Related papers  (2023-10-04T09:39:05Z)
• Stochastic Interpolants: A Unifying Framework for Flows and Diffusions [16.95541777254722]
  A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & VandenEijnden (2023), enabling the use of a broad class of continuous-time processes called stochastic interpolants' These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way.
  arXiv  Detail & Related papers  (2023-03-15T17:43:42Z)
• Building Normalizing Flows with Stochastic Interpolants [11.22149158986164]
  A simple generative quadratic model based on a continuous-time normalizing flow between any pair of base and target distributions is proposed. The velocity field of this flow is inferred from the probability current of a time-dependent distribution that interpolates between the base and the target in finite time.
  arXiv  Detail & Related papers  (2022-09-30T16:30:31Z)
• Manifold Interpolating Optimal-Transport Flows for Trajectory Inference [64.94020639760026]
  We present a method called Manifold Interpolating Optimal-Transport Flow (MIOFlow) MIOFlow learns, continuous population dynamics from static snapshot samples taken at sporadic timepoints. We evaluate our method on simulated data with bifurcations and merges, as well as scRNA-seq data from embryoid body differentiation, and acute myeloid leukemia treatment.
  arXiv  Detail & Related papers  (2022-06-29T22:19:03Z)
• The Boomerang Sampler [4.588028371034406]
  This paper introduces the Boomerang Sampler as a novel class of continuous-time non-reversible Markov chain Monte Carlo algorithms. We demonstrate that the method is easy to implement and demonstrate empirically that it can out-perform existing benchmark piecewise deterministic Markov processes.
  arXiv  Detail & Related papers  (2020-06-24T14:52:22Z)

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.