Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans

2025-01-28

AI Summary

This work addresses the modeling and learning of velocity fields governing data distribution evolution in flow matching, aiming to unify transport planning, Markov kernels, and stochastic process paradigms. Methodologically, it establishes the first theoretical equivalence framework for velocity fields characterizing absolutely continuous Wasserstein curves across these three constructions; introduces the conditional Wasserstein distance as a novel metric for Bayesian inverse problems; and unifies the geometric interpretations of continuous normalizing flows and score matching. Leveraging tools from Wasserstein geometry, optimal transport, and stochastic differential equations, the paper rigorously proves the intrinsic consistency of multiple velocity field learning approaches, thereby strengthening the mathematical foundations of flow matching. Experiments demonstrate that the proposed framework effectively generates high-fidelity conditional distributions in Bayesian inverse problems.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Data State Evolution
• Bayesian Inverse Problems
• Continuous Regularization

Innovation

• Flow Matching
• Bayesian Inverse Problems
• Continuous Regularization Flows