Topological Flow Matching

2026-06-14

AI Summary

This work addresses a key limitation of standard flow matching methods, which treat structured signals—such as fMRI data on brain graphs—as points in Euclidean space and thereby ignore the underlying topological structure of their domain. To overcome this, the authors propose Topological Flow Matching, the first approach to explicitly incorporate domain topology into the flow matching framework. By introducing a graph Laplacian–based drift term into the reference process, the method constructs a topology-aware deterministic generative trajectory that is both theoretically grounded and readily integrable into existing pipelines. Built upon a degenerate formulation of the Schrödinger bridge problem, it enables stable generation without requiring stochastic simulation. Empirical evaluations across diverse structured datasets—including brain fMRI, ocean currents, seismic events, and traffic flows—demonstrate substantial improvements over conventional flow matching approaches that neglect topological information.

Abstract

Flow matching is a powerful generative modeling framework, valued for its simplicity and strong empirical performance. However, its standard formulation treats signals on structured spaces, such as fMRI data on brain graphs, as points in Euclidean space, overlooking the rich topological features of their domains. To address this, we introduce topological flow matching, a topology-aware generalization of flow matching. We interpret flow matching as a framework for solving a degenerate Schrödinger bridge problem and inject topological information by augmenting the reference process with a Laplacian-derived drift. This principled modification captures the structure of the underlying domain while preserving the desirable properties of flow matching: a stable, simulation-free objective and deterministic sample paths. As a result, our framework serves as a drop-in replacement for standard flow matching. We demonstrate its effectiveness on diverse structured datasets, including brain fMRIs, ocean currents, seismic events, and traffic flows.

Problem

• flow matching
• topological features
• structured spaces
• graph-structured data
• generative modeling

Innovation

• topological flow matching
• Schrödinger bridge
• Laplacian drift
• structured data generation
• non-Euclidean domains