Rigid Invariant Sliced Wasserstein via Independent Embeddings

Zakk Heile, Peilin He, Jayson Tran, Alice Wang, Shrikant Chand
Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling, PMLR 326:321-355, 2026.

Abstract

Comparing probability measures modulo unknown rigid transformations is a central challenge in geometric data analysis. Classical optimal transport (OT) distances, including Wasserstein and sliced Wasserstein, are sensitive to rotations and reflections, whereas Gromov-Wasserstein (GW) distances are invariant to isometries but computationally prohibitive for large datasets. We introduce Rigid-Invariant Sliced Wasserstein via Independent Embeddings (RISWIE), a scalable distance that combines the invariance of NP-hard approaches with the efficiency of projection-based OT. RISWIE utilizes data-adaptive bases and matches optimal signed permutations along axes according to distributional similarity to achieve rigid invariance with near-linear complexity in the sample size. We prove bounds relating RISWIE to GW in special cases and demonstrate dimension-independent statistical stability. Our experiments on cellular imaging and 3D human meshes demonstrate that RISWIE outperforms GW in clustering tasks and discriminative capability while significantly reducing runtime.

Cite this Paper

BibTeX
@InProceedings{pmlr-v326-heile26a, title = {Rigid Invariant Sliced Wasserstein via Independent Embeddings}, author = {Heile, Zakk and He, Peilin and Tran, Jayson and Wang, Alice and Chand, Shrikant}, booktitle = {Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling}, pages = {321--355}, year = {2026}, editor = {Pouplin, Alison and Vadgama, Sharvaree and Bekkers, Erik and Kaba, Sékou-Oumar and Lawrence, Hannah and Lecha, Manuel and Baker, Elizabeth and Suk, Julian and Walters, Robin and Tomczak, Jakub and Jegelka, Stefanie}, volume = {326}, series = {Proceedings of Machine Learning Research}, month = {26 Apr}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v326/main/assets/heile26a/heile26a.pdf}, url = {https://proceedings.mlr.press/v326/heile26a.html}, abstract = {Comparing probability measures modulo unknown rigid transformations is a central challenge in geometric data analysis. Classical optimal transport (OT) distances, including Wasserstein and sliced Wasserstein, are sensitive to rotations and reflections, whereas Gromov-Wasserstein (GW) distances are invariant to isometries but computationally prohibitive for large datasets. We introduce Rigid-Invariant Sliced Wasserstein via Independent Embeddings (RISWIE), a scalable distance that combines the invariance of NP-hard approaches with the efficiency of projection-based OT. RISWIE utilizes data-adaptive bases and matches optimal signed permutations along axes according to distributional similarity to achieve rigid invariance with near-linear complexity in the sample size. We prove bounds relating RISWIE to GW in special cases and demonstrate dimension-independent statistical stability. Our experiments on cellular imaging and 3D human meshes demonstrate that RISWIE outperforms GW in clustering tasks and discriminative capability while significantly reducing runtime.} }
Endnote
%0 Conference Paper %T Rigid Invariant Sliced Wasserstein via Independent Embeddings %A Zakk Heile %A Peilin He %A Jayson Tran %A Alice Wang %A Shrikant Chand %B Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling %C Proceedings of Machine Learning Research %D 2026 %E Alison Pouplin %E Sharvaree Vadgama %E Erik Bekkers %E Sékou-Oumar Kaba %E Hannah Lawrence %E Manuel Lecha %E Elizabeth Baker %E Julian Suk %E Robin Walters %E Jakub Tomczak %E Stefanie Jegelka %F pmlr-v326-heile26a %I PMLR %P 321--355 %U https://proceedings.mlr.press/v326/heile26a.html %V 326 %X Comparing probability measures modulo unknown rigid transformations is a central challenge in geometric data analysis. Classical optimal transport (OT) distances, including Wasserstein and sliced Wasserstein, are sensitive to rotations and reflections, whereas Gromov-Wasserstein (GW) distances are invariant to isometries but computationally prohibitive for large datasets. We introduce Rigid-Invariant Sliced Wasserstein via Independent Embeddings (RISWIE), a scalable distance that combines the invariance of NP-hard approaches with the efficiency of projection-based OT. RISWIE utilizes data-adaptive bases and matches optimal signed permutations along axes according to distributional similarity to achieve rigid invariance with near-linear complexity in the sample size. We prove bounds relating RISWIE to GW in special cases and demonstrate dimension-independent statistical stability. Our experiments on cellular imaging and 3D human meshes demonstrate that RISWIE outperforms GW in clustering tasks and discriminative capability while significantly reducing runtime.
APA
Heile, Z., He, P., Tran, J., Wang, A. & Chand, S.. (2026). Rigid Invariant Sliced Wasserstein via Independent Embeddings. Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling, in Proceedings of Machine Learning Research 326:321-355 Available from https://proceedings.mlr.press/v326/heile26a.html.

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