Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates

Rémi Leluc, Aymeric Dieuleveut, François Portier, Johan Segers, Aigerim Zhuman
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:27191-27214, 2024.

Abstract

The Sliced-Wasserstein (SW) distance between probability measures is defined as the average of the Wasserstein distances resulting for the associated one-dimensional projections. As a consequence, the SW distance can be written as an integral with respect to the uniform measure on the sphere and the Monte Carlo framework can be employed for calculating the SW distance. Spherical harmonics are polynomials on the sphere that form an orthonormal basis of the set of square-integrable functions on the sphere. Putting these two facts together, a new Monte Carlo method, hereby referred to as Spherical Harmonics Control Variates (SHCV), is proposed for approximating the SW distance using spherical harmonics as control variates. The resulting approach is shown to have good theoretical properties, e.g., a no-error property for Gaussian measures under a certain form of linear dependency between the variables. Moreover, an improved rate of convergence, compared to Monte Carlo, is established for general measures. The convergence analysis relies on the Lipschitz property associated to the SW integrand. Several numerical experiments demonstrate the superior performance of SHCV against state-of-the-art methods for SW distance computation.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-leluc24a, title = {Sliced-{W}asserstein Estimation with Spherical Harmonics as Control Variates}, author = {Leluc, R\'{e}mi and Dieuleveut, Aymeric and Portier, Fran\c{c}ois and Segers, Johan and Zhuman, Aigerim}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {27191--27214}, 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/leluc24a/leluc24a.pdf}, url = {https://proceedings.mlr.press/v235/leluc24a.html}, abstract = {The Sliced-Wasserstein (SW) distance between probability measures is defined as the average of the Wasserstein distances resulting for the associated one-dimensional projections. As a consequence, the SW distance can be written as an integral with respect to the uniform measure on the sphere and the Monte Carlo framework can be employed for calculating the SW distance. Spherical harmonics are polynomials on the sphere that form an orthonormal basis of the set of square-integrable functions on the sphere. Putting these two facts together, a new Monte Carlo method, hereby referred to as Spherical Harmonics Control Variates (SHCV), is proposed for approximating the SW distance using spherical harmonics as control variates. The resulting approach is shown to have good theoretical properties, e.g., a no-error property for Gaussian measures under a certain form of linear dependency between the variables. Moreover, an improved rate of convergence, compared to Monte Carlo, is established for general measures. The convergence analysis relies on the Lipschitz property associated to the SW integrand. Several numerical experiments demonstrate the superior performance of SHCV against state-of-the-art methods for SW distance computation.} }
Endnote
%0 Conference Paper %T Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates %A Rémi Leluc %A Aymeric Dieuleveut %A François Portier %A Johan Segers %A Aigerim Zhuman %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-leluc24a %I PMLR %P 27191--27214 %U https://proceedings.mlr.press/v235/leluc24a.html %V 235 %X The Sliced-Wasserstein (SW) distance between probability measures is defined as the average of the Wasserstein distances resulting for the associated one-dimensional projections. As a consequence, the SW distance can be written as an integral with respect to the uniform measure on the sphere and the Monte Carlo framework can be employed for calculating the SW distance. Spherical harmonics are polynomials on the sphere that form an orthonormal basis of the set of square-integrable functions on the sphere. Putting these two facts together, a new Monte Carlo method, hereby referred to as Spherical Harmonics Control Variates (SHCV), is proposed for approximating the SW distance using spherical harmonics as control variates. The resulting approach is shown to have good theoretical properties, e.g., a no-error property for Gaussian measures under a certain form of linear dependency between the variables. Moreover, an improved rate of convergence, compared to Monte Carlo, is established for general measures. The convergence analysis relies on the Lipschitz property associated to the SW integrand. Several numerical experiments demonstrate the superior performance of SHCV against state-of-the-art methods for SW distance computation.
APA
Leluc, R., Dieuleveut, A., Portier, F., Segers, J. & Zhuman, A.. (2024). Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:27191-27214 Available from https://proceedings.mlr.press/v235/leluc24a.html.

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