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Generalized Wasserstein Flow Matching: Nested Couplings for Point Cloud Generation

  • Moritz Piening
  • Richard Duong
  • Gabriele Steidl
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Abstract

We formulate flow matching on probability measures over probability measures. Directly from the Wasserstein-on-Wasserstein (WoW) continuity equation, we show that admissible nested outer and inner couplings induce a valid metameasure flow and non-local velocity field. Validity requires neither optimal plans nor a smooth manifold. Practically, this opens up the design space of multisample flow matching: exact Wasserstein plans are distinguished by energy and straightness, while sliced and lazy linear plans yield scalable alternatives. Point-cloud experiments, including controlled Bures-Wasserstein and Euclidean minibatch baselines, demonstrate the resulting geometry-cost trade-off.

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Cite

BibTeXRIS
BibTeX
@misc{piening2026generalized,
  title = {{Generalized Wasserstein Flow Matching: Nested Couplings for Point Cloud Generation}},
  author = {Moritz Piening and Richard Duong and Gabriele Steidl},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.08424},
  eprint = {2605.08424},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.08424},
}
RIS
TY  - UNPB
TI  - Generalized Wasserstein Flow Matching: Nested Couplings for Point Cloud Generation
AU  - Moritz Piening
AU  - Richard Duong
AU  - Gabriele Steidl
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2605.08424
UR  - https://arxiv.org/abs/2605.08424
AB  - We formulate flow matching on probability measures over probability measures. Directly from the Wasserstein-on-Wasserstein (WoW) continuity equation, we show that admissible nested outer and inner couplings induce a valid metameasure flow and non-local velocity field. Validity requires neither optimal plans nor a smooth manifold. Practically, this opens up the design space of multisample flow matching: exact Wasserstein plans are distinguished by energy and straightness, while sliced and lazy linear plans yield scalable alternatives. Point-cloud experiments, including controlled Bures-Wasserstein and Euclidean minibatch baselines, demonstrate the resulting geometry-cost trade-off.
ER  -