Stereographic Spherical Sliced Wasserstein Distances

Huy Tran, Yikun Bai, Abihith Kothapalli, Ashkan Shahbazi, Xinran Liu, Rocio P Diaz Martin, Soheil Kolouri
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:48494-48564, 2024.

Abstract

Comparing spherical probability distributions is of great interest in various fields, including geology, medical domains, computer vision, and deep representation learning. The utility of optimal transport-based distances, such as the Wasserstein distance, for comparing probability measures has spurred active research in developing computationally efficient variations of these distances for spherical probability measures. This paper introduces a high-speed and highly parallelizable distance for comparing spherical measures using the stereographic projection and the generalized Radon transform, which we refer to as the Stereographic Spherical Sliced Wasserstein (S3W) distance. We carefully address the distance distortion caused by the stereographic projection and provide an extensive theoretical analysis of our proposed metric and its rotationally invariant variation. Finally, we evaluate the performance of the proposed metrics and compare them with recent baselines in terms of both speed and accuracy through a wide range of numerical studies, including gradient flows and self-supervised learning. Our code is available at https://github.com/mint-vu/s3wd.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-tran24a, title = {Stereographic Spherical Sliced {W}asserstein Distances}, author = {Tran, Huy and Bai, Yikun and Kothapalli, Abihith and Shahbazi, Ashkan and Liu, Xinran and Diaz Martin, Rocio P and Kolouri, Soheil}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {48494--48564}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/tran24a/tran24a.pdf}, url = {https://proceedings.mlr.press/v235/tran24a.html}, abstract = {Comparing spherical probability distributions is of great interest in various fields, including geology, medical domains, computer vision, and deep representation learning. The utility of optimal transport-based distances, such as the Wasserstein distance, for comparing probability measures has spurred active research in developing computationally efficient variations of these distances for spherical probability measures. This paper introduces a high-speed and highly parallelizable distance for comparing spherical measures using the stereographic projection and the generalized Radon transform, which we refer to as the Stereographic Spherical Sliced Wasserstein (S3W) distance. We carefully address the distance distortion caused by the stereographic projection and provide an extensive theoretical analysis of our proposed metric and its rotationally invariant variation. Finally, we evaluate the performance of the proposed metrics and compare them with recent baselines in terms of both speed and accuracy through a wide range of numerical studies, including gradient flows and self-supervised learning. Our code is available at https://github.com/mint-vu/s3wd.} }
Endnote
%0 Conference Paper %T Stereographic Spherical Sliced Wasserstein Distances %A Huy Tran %A Yikun Bai %A Abihith Kothapalli %A Ashkan Shahbazi %A Xinran Liu %A Rocio P Diaz Martin %A Soheil Kolouri %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-tran24a %I PMLR %P 48494--48564 %U https://proceedings.mlr.press/v235/tran24a.html %V 235 %X Comparing spherical probability distributions is of great interest in various fields, including geology, medical domains, computer vision, and deep representation learning. The utility of optimal transport-based distances, such as the Wasserstein distance, for comparing probability measures has spurred active research in developing computationally efficient variations of these distances for spherical probability measures. This paper introduces a high-speed and highly parallelizable distance for comparing spherical measures using the stereographic projection and the generalized Radon transform, which we refer to as the Stereographic Spherical Sliced Wasserstein (S3W) distance. We carefully address the distance distortion caused by the stereographic projection and provide an extensive theoretical analysis of our proposed metric and its rotationally invariant variation. Finally, we evaluate the performance of the proposed metrics and compare them with recent baselines in terms of both speed and accuracy through a wide range of numerical studies, including gradient flows and self-supervised learning. Our code is available at https://github.com/mint-vu/s3wd.
APA
Tran, H., Bai, Y., Kothapalli, A., Shahbazi, A., Liu, X., Diaz Martin, R.P. & Kolouri, S.. (2024). Stereographic Spherical Sliced Wasserstein Distances. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:48494-48564 Available from https://proceedings.mlr.press/v235/tran24a.html.

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