The Shape of Flows: Topological Regularization of Vector Fields in Flow Matching for Better Diversity and Fidelity.

Flow Matching models have become the de facto paradigm for the generation of continuous data. Despite impressive results, conditional sampling from these models implies a fidelity/diversity tradeoff, which greatly impacts their applicability. To address this issue, we propose the use of persistent homology to identify relevant topological features both at the pixel level and at the sample level, from which we derive two training objectives. On the one hand, we use filtrations on cubical complexes to align the topological persistence of pixel velocity sublevel sets with that of the ground truths; on the other hand, we employ filtrations on Vietoris-Rips complexes to ensure the topology of the generated distribution approximates that of the training set. We demonstrate these approaches improve both the diversity and fidelity of generations, showing the promise of persistent homology and geometric invariants to improve the quality of Flow Matching models.