Tree-Sliced Wasserstein Distance: A Geometric Perspective

Hoang V. Tran, Huyen Trang Pham, Tho Tran Huu, Minh-Khoi Nguyen-Nhat, Thanh Chu, Tam Le, Tan Minh Nguyen
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:59969-60000, 2025.

Abstract

Many variants of Optimal Transport (OT) have been developed to address its heavy computation. Among them, notably, Sliced Wasserstein (SW) is widely used for application domains by projecting the OT problem onto one-dimensional lines, and leveraging the closed-form expression of the univariate OT to reduce the computational burden. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. To mitigate this issue, in this work, we propose to replace one-dimensional lines with a more intricate structure, called tree systems. This structure is metrizable by a tree metric, which yields a closed-form expression for OT problems on tree systems. We provide an extensive theoretical analysis to formally define tree systems with their topological properties, introduce the concept of splitting maps, which operate as the projection mechanism onto these structures, then finally propose a novel variant of Radon transform for tree systems and verify its injectivity. This framework leads to an efficient metric between measures, termed Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL). By conducting a variety of experiments on gradient flows, image style transfer, and generative models, we illustrate that our proposed approach performs favorably compared to SW and its variants.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-tran25b, title = {Tree-Sliced {W}asserstein Distance: A Geometric Perspective}, author = {Tran, Hoang V. and Pham, Huyen Trang and Huu, Tho Tran and Nguyen-Nhat, Minh-Khoi and Chu, Thanh and Le, Tam and Nguyen, Tan Minh}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {59969--60000}, 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/tran25b/tran25b.pdf}, url = {https://proceedings.mlr.press/v267/tran25b.html}, abstract = {Many variants of Optimal Transport (OT) have been developed to address its heavy computation. Among them, notably, Sliced Wasserstein (SW) is widely used for application domains by projecting the OT problem onto one-dimensional lines, and leveraging the closed-form expression of the univariate OT to reduce the computational burden. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. To mitigate this issue, in this work, we propose to replace one-dimensional lines with a more intricate structure, called tree systems. This structure is metrizable by a tree metric, which yields a closed-form expression for OT problems on tree systems. We provide an extensive theoretical analysis to formally define tree systems with their topological properties, introduce the concept of splitting maps, which operate as the projection mechanism onto these structures, then finally propose a novel variant of Radon transform for tree systems and verify its injectivity. This framework leads to an efficient metric between measures, termed Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL). By conducting a variety of experiments on gradient flows, image style transfer, and generative models, we illustrate that our proposed approach performs favorably compared to SW and its variants.} }
Endnote
%0 Conference Paper %T Tree-Sliced Wasserstein Distance: A Geometric Perspective %A Hoang V. Tran %A Huyen Trang Pham %A Tho Tran Huu %A Minh-Khoi Nguyen-Nhat %A Thanh Chu %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-tran25b %I PMLR %P 59969--60000 %U https://proceedings.mlr.press/v267/tran25b.html %V 267 %X Many variants of Optimal Transport (OT) have been developed to address its heavy computation. Among them, notably, Sliced Wasserstein (SW) is widely used for application domains by projecting the OT problem onto one-dimensional lines, and leveraging the closed-form expression of the univariate OT to reduce the computational burden. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. To mitigate this issue, in this work, we propose to replace one-dimensional lines with a more intricate structure, called tree systems. This structure is metrizable by a tree metric, which yields a closed-form expression for OT problems on tree systems. We provide an extensive theoretical analysis to formally define tree systems with their topological properties, introduce the concept of splitting maps, which operate as the projection mechanism onto these structures, then finally propose a novel variant of Radon transform for tree systems and verify its injectivity. This framework leads to an efficient metric between measures, termed Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL). By conducting a variety of experiments on gradient flows, image style transfer, and generative models, we illustrate that our proposed approach performs favorably compared to SW and its variants.
APA
Tran, H.V., Pham, H.T., Huu, T.T., Nguyen-Nhat, M., Chu, T., Le, T. & Nguyen, T.M.. (2025). Tree-Sliced Wasserstein Distance: A Geometric Perspective. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:59969-60000 Available from https://proceedings.mlr.press/v267/tran25b.html.

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