Sinkhorn Flow as Mirror Flow: A Continuous-Time Framework for Generalizing the Sinkhorn Algorithm

Mohammad Reza Karimi, Ya-Ping Hsieh, Andreas Krause
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:4186-4194, 2024.

Abstract

Many problems in machine learning can be formulated as solving entropy-regularized optimal transport on the space of probability measures. The canonical approach involves the Sinkhorn iterates, renowned for their rich mathematical properties. Recently, the Sinkhorn algorithm has been recast within the mirror descent framework, thus benefiting from classical optimization theory insights. Here, we build upon this result by introducing a continuous-time analogue of the Sinkhorn algorithm. This perspective allows us to derive novel variants of Sinkhorn schemes that are robust to noise and bias. Moreover, our continuous-time dynamics offers a unified perspective on several recently discovered dynamics in machine learning and mathematics, such as the "Wasserstein mirror flow" of Deb et al. (2023) or the "mean-field Schrödinger equation" of Claisse et al. (2023).

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-reza-karimi24a, title = {Sinkhorn Flow as Mirror Flow: A Continuous-Time Framework for Generalizing the {S}inkhorn Algorithm}, author = {Reza Karimi, Mohammad and Hsieh, Ya-Ping and Krause, Andreas}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {4186--4194}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/reza-karimi24a/reza-karimi24a.pdf}, url = {https://proceedings.mlr.press/v238/reza-karimi24a.html}, abstract = {Many problems in machine learning can be formulated as solving entropy-regularized optimal transport on the space of probability measures. The canonical approach involves the Sinkhorn iterates, renowned for their rich mathematical properties. Recently, the Sinkhorn algorithm has been recast within the mirror descent framework, thus benefiting from classical optimization theory insights. Here, we build upon this result by introducing a continuous-time analogue of the Sinkhorn algorithm. This perspective allows us to derive novel variants of Sinkhorn schemes that are robust to noise and bias. Moreover, our continuous-time dynamics offers a unified perspective on several recently discovered dynamics in machine learning and mathematics, such as the "Wasserstein mirror flow" of Deb et al. (2023) or the "mean-field Schrödinger equation" of Claisse et al. (2023).} }
Endnote
%0 Conference Paper %T Sinkhorn Flow as Mirror Flow: A Continuous-Time Framework for Generalizing the Sinkhorn Algorithm %A Mohammad Reza Karimi %A Ya-Ping Hsieh %A Andreas Krause %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-reza-karimi24a %I PMLR %P 4186--4194 %U https://proceedings.mlr.press/v238/reza-karimi24a.html %V 238 %X Many problems in machine learning can be formulated as solving entropy-regularized optimal transport on the space of probability measures. The canonical approach involves the Sinkhorn iterates, renowned for their rich mathematical properties. Recently, the Sinkhorn algorithm has been recast within the mirror descent framework, thus benefiting from classical optimization theory insights. Here, we build upon this result by introducing a continuous-time analogue of the Sinkhorn algorithm. This perspective allows us to derive novel variants of Sinkhorn schemes that are robust to noise and bias. Moreover, our continuous-time dynamics offers a unified perspective on several recently discovered dynamics in machine learning and mathematics, such as the "Wasserstein mirror flow" of Deb et al. (2023) or the "mean-field Schrödinger equation" of Claisse et al. (2023).
APA
Reza Karimi, M., Hsieh, Y. & Krause, A.. (2024). Sinkhorn Flow as Mirror Flow: A Continuous-Time Framework for Generalizing the Sinkhorn Algorithm. Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:4186-4194 Available from https://proceedings.mlr.press/v238/reza-karimi24a.html.

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