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This research extends the flow matching generative modeling framework by integrating topological constraints to better handle structured data like brain fMRI scans and ocean currents. By augmenting the reference process with a Laplacian-derived drift, it captures domain-specific geometry while maintaining stable, deterministic sampling without simulation overhead. The method offers a principled drop-in replacement for standard approaches in scenarios where Euclidean assumptions fail on non-Euclidean manifolds.
Topological Flow Matching
arXiv:2606.15897v1 Announce Type: new 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 Schr\"odinger 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.
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.
Accepted at ICLR 2026. 26 pages, 24 figures. Code: this https URL
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Machine Learning (stat.ML)
arXiv:2606.15897 [cs.LG]
(or arXiv:2606.15897v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.15897 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 256 words
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