Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis

Christophe Vauthier, Anna Korba, Quentin Mérigot
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:61071-61107, 2025.

Abstract

In this paper, we investigate the properties of the Sliced Wasserstein Distance (SW) when employed as an objective functional. The SW metric has gained significant interest in the optimal transport and machine learning literature, due to its ability to capture intricate geometric properties of probability distributions while remaining computationally tractable, making it a valuable tool for various applications, including generative modeling and domain adaptation. Our study aims to provide a rigorous analysis of the critical points arising from the optimization of the SW objective. By computing explicit perturbations, we establish that stable critical points of SW cannot concentrate on segments. This stability analysis is crucial for understanding the behaviour of optimization algorithms for models trained using the SW objective. Furthermore, we investigate the properties of the SW objective, shedding light on the existence and convergence behavior of critical points. We illustrate our theoretical results through numerical experiments.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-vauthier25a, title = {Towards Understanding Gradient Dynamics of the Sliced-{W}asserstein Distance via Critical Point Analysis}, author = {Vauthier, Christophe and Korba, Anna and M\'{e}rigot, Quentin}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {61071--61107}, 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/vauthier25a/vauthier25a.pdf}, url = {https://proceedings.mlr.press/v267/vauthier25a.html}, abstract = {In this paper, we investigate the properties of the Sliced Wasserstein Distance (SW) when employed as an objective functional. The SW metric has gained significant interest in the optimal transport and machine learning literature, due to its ability to capture intricate geometric properties of probability distributions while remaining computationally tractable, making it a valuable tool for various applications, including generative modeling and domain adaptation. Our study aims to provide a rigorous analysis of the critical points arising from the optimization of the SW objective. By computing explicit perturbations, we establish that stable critical points of SW cannot concentrate on segments. This stability analysis is crucial for understanding the behaviour of optimization algorithms for models trained using the SW objective. Furthermore, we investigate the properties of the SW objective, shedding light on the existence and convergence behavior of critical points. We illustrate our theoretical results through numerical experiments.} }
Endnote
%0 Conference Paper %T Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis %A Christophe Vauthier %A Anna Korba %A Quentin Mérigot %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-vauthier25a %I PMLR %P 61071--61107 %U https://proceedings.mlr.press/v267/vauthier25a.html %V 267 %X In this paper, we investigate the properties of the Sliced Wasserstein Distance (SW) when employed as an objective functional. The SW metric has gained significant interest in the optimal transport and machine learning literature, due to its ability to capture intricate geometric properties of probability distributions while remaining computationally tractable, making it a valuable tool for various applications, including generative modeling and domain adaptation. Our study aims to provide a rigorous analysis of the critical points arising from the optimization of the SW objective. By computing explicit perturbations, we establish that stable critical points of SW cannot concentrate on segments. This stability analysis is crucial for understanding the behaviour of optimization algorithms for models trained using the SW objective. Furthermore, we investigate the properties of the SW objective, shedding light on the existence and convergence behavior of critical points. We illustrate our theoretical results through numerical experiments.
APA
Vauthier, C., Korba, A. & Mérigot, Q.. (2025). Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:61071-61107 Available from https://proceedings.mlr.press/v267/vauthier25a.html.

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