Stereographic Spherical Sliced Wasserstein Distances
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916280868536320 |
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| author | Tran, Huy Bai, Yikun Kothapalli, Abihith Shahbazi, Ashkan Liu, Xinran Martin, Rocio Diaz Kolouri, Soheil |
| author_facet | Tran, Huy Bai, Yikun Kothapalli, Abihith Shahbazi, Ashkan Liu, Xinran Martin, Rocio Diaz Kolouri, Soheil |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02345 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stereographic Spherical Sliced Wasserstein Distances Tran, Huy Bai, Yikun Kothapalli, Abihith Shahbazi, Ashkan Liu, Xinran Martin, Rocio Diaz Kolouri, Soheil Machine Learning Artificial Intelligence Computer Vision and Pattern Recognition 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. |
| title | Stereographic Spherical Sliced Wasserstein Distances |
| topic | Machine Learning Artificial Intelligence Computer Vision and Pattern Recognition |
| url | https://arxiv.org/abs/2402.02345 |