Tree-Sliced Wasserstein Distance with Nonlinear Projection

Thanh Tran, Hoang V. Tran, Thanh Chu, Huyen Trang Pham, Laurent El Ghaoui, Tam Le, Tan Minh Nguyen
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:60001-60033, 2025.

Abstract

Tree-Sliced methods have recently emerged as an alternative to the traditional Sliced Wasserstein (SW) distance, replacing one-dimensional lines with tree-based metric spaces and incorporating a splitting mechanism for projecting measures. This approach enhances the ability to capture the topological structures of integration domains in Sliced Optimal Transport while maintaining low computational costs. Building on this foundation, we propose a novel nonlinear projectional framework for the Tree-Sliced Wasserstein (TSW) distance, substituting the linear projections in earlier versions with general projections, while ensuring the injectivity of the associated Radon Transform and preserving the well-definedness of the resulting metric. By designing appropriate projections, we construct efficient metrics for measures on both Euclidean spaces and spheres. Finally, we validate our proposed metric through extensive numerical experiments for Euclidean and spherical datasets. Applications include gradient flows, self-supervised learning, and generative models, where our methods demonstrate significant improvements over recent SW and TSW variants.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-tran25c, title = {Tree-Sliced {W}asserstein Distance with Nonlinear Projection}, author = {Tran, Thanh and Tran, Hoang V. and Chu, Thanh and Pham, Huyen Trang and El Ghaoui, Laurent and Le, Tam and Nguyen, Tan Minh}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {60001--60033}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/tran25c/tran25c.pdf}, url = {https://proceedings.mlr.press/v267/tran25c.html}, abstract = {Tree-Sliced methods have recently emerged as an alternative to the traditional Sliced Wasserstein (SW) distance, replacing one-dimensional lines with tree-based metric spaces and incorporating a splitting mechanism for projecting measures. This approach enhances the ability to capture the topological structures of integration domains in Sliced Optimal Transport while maintaining low computational costs. Building on this foundation, we propose a novel nonlinear projectional framework for the Tree-Sliced Wasserstein (TSW) distance, substituting the linear projections in earlier versions with general projections, while ensuring the injectivity of the associated Radon Transform and preserving the well-definedness of the resulting metric. By designing appropriate projections, we construct efficient metrics for measures on both Euclidean spaces and spheres. Finally, we validate our proposed metric through extensive numerical experiments for Euclidean and spherical datasets. Applications include gradient flows, self-supervised learning, and generative models, where our methods demonstrate significant improvements over recent SW and TSW variants.} }
Endnote
%0 Conference Paper %T Tree-Sliced Wasserstein Distance with Nonlinear Projection %A Thanh Tran %A Hoang V. Tran %A Thanh Chu %A Huyen Trang Pham %A Laurent El Ghaoui %A Tam Le %A Tan Minh Nguyen %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-tran25c %I PMLR %P 60001--60033 %U https://proceedings.mlr.press/v267/tran25c.html %V 267 %X Tree-Sliced methods have recently emerged as an alternative to the traditional Sliced Wasserstein (SW) distance, replacing one-dimensional lines with tree-based metric spaces and incorporating a splitting mechanism for projecting measures. This approach enhances the ability to capture the topological structures of integration domains in Sliced Optimal Transport while maintaining low computational costs. Building on this foundation, we propose a novel nonlinear projectional framework for the Tree-Sliced Wasserstein (TSW) distance, substituting the linear projections in earlier versions with general projections, while ensuring the injectivity of the associated Radon Transform and preserving the well-definedness of the resulting metric. By designing appropriate projections, we construct efficient metrics for measures on both Euclidean spaces and spheres. Finally, we validate our proposed metric through extensive numerical experiments for Euclidean and spherical datasets. Applications include gradient flows, self-supervised learning, and generative models, where our methods demonstrate significant improvements over recent SW and TSW variants.
APA
Tran, T., Tran, H.V., Chu, T., Pham, H.T., El Ghaoui, L., Le, T. & Nguyen, T.M.. (2025). Tree-Sliced Wasserstein Distance with Nonlinear Projection. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:60001-60033 Available from https://proceedings.mlr.press/v267/tran25c.html.

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