Research map / Probabilistic and scientific modeling / Variational and generative sampling
Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Extends flow matching to Wasserstein-on-Wasserstein space of metameasures, using outer/inner transport plans with sliced and lazy linear Wasserstein approximations for point cloud and set generation.
Problem
Flow matching over probability measures over probability measures (metameasures) requires costly nested optimal transport couplings, making training on point clouds and sets expensive.
Approach
Lift flow matching to the WoW space by coupling outer plans between empirical measures with inner plans between points; derive a non-local velocity field as minimizer of a flow matching loss, and approximate the outer coupling with sliced Wasserstein or a lazy linearized Wasserstein distance that needs only one OT solve per sample.
Key results
- On MNIST point clouds with barycentric noise, the linearized outer and inner coupling (LLW) reaches OT-NNA 0.57 at Euler-125, comparable to the full WoW coupling (0.58) at 3-4h training versus 9-10h.
- On ShapeNet airplanes (N=512), the sliced outer coupling achieves the best OT-NNA of 0.55 at Euler-125 versus 0.64 for independent coupling.
- On MNIST to USPS transport with N=256, the WoW OT coupling gives the best Chamfer-NNA at Euler-5 (0.64 vs 1.00 for independent), while sliced coupling gives 0.94 at Euler-5.
Surprising findings
- With enough Euler steps, the independent coupling becomes competitive with OT-based couplings on the MNIST to USPS task.
Limitations
- WoW setup not evaluated on high-resolution ShapeNet due to computational cost
- Lazy linearization relies on source metameasure concentrated around a barycenter reference, with theory only holding with high probability
- Experiments are limited to 2D and 3D point clouds and sets
Experimental setup
Datasets: MNIST ShapeNet USPS
Benchmarks: MNIST point clouds ShapeNet airplanes MNIST to USPS transport
Metrics: Chamfer-NNA OT-NNA training time
Baselines: Wasserstein flow matching (independent outer coupling) WoW OT coupling independent coupling
Builds on: flow matching Wasserstein flow matching linearized optimal transport sliced Wasserstein distance Riemannian flow matching
Closest papers by meaning
- Wasserstein Flow Matching: Generative Modeling Over Families of Distributions (ICML 2025 · similarity 0.91)
- Optimal Flow Matching: Learning Straight Trajectories in Just One Step (NeurIPS 2024 · similarity 0.89)
- On the Relation between Rectified Flows and Optimal Transport (NeurIPS 2025 · similarity 0.89)
- Beckmann Transport Models: From Autonomous Flows to One-Step Maps (NeurIPS 2026 · similarity 0.88)
- Scalable Wasserstein Gradient Flow for Generative Modeling through Unbalanced Optimal Transport (ICML 2024 · similarity 0.88)
- Flow matching achieves almost minimax optimal convergence (ICLR 2025 · similarity 0.88)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.