Debiased Sinkhorn barycenters

Hicham Janati, Marco Cuturi, Alexandre Gramfort
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:4692-4701, 2020.

Abstract

Entropy regularization in optimal transport (OT) has been the driver of many recent interests for Wasserstein metrics and barycenters in machine learning. It allows to keep the appealing geometrical properties of the unregularized Wasserstein distance while having a significantly lower complexity thanks to Sinkhorn’s algorithm. However, entropy brings some inherent smoothing bias, resulting for example in blurred barycenters. This side effect has prompted an increasing temptation in the community to settle for a slower algorithm such as log-domain stabilized Sinkhorn which breaks the parallel structure that can be leveraged on GPUs, or even go back to unregularized OT. Here we show how this bias is tightly linked to the reference measure that defines the entropy regularizer and propose debiased Sinkhorn barycenters that preserve the best of worlds: fast Sinkhorn-like iterations without entropy smoothing. Theoretically, we prove that this debiasing is perfect for Gaussian distributions with equal variance. Empirically, we illustrate the reduced blurring and the computational advantage.

Cite this Paper


BibTeX
@InProceedings{pmlr-v119-janati20a, title = {Debiased {S}inkhorn barycenters}, author = {Janati, Hicham and Cuturi, Marco and Gramfort, Alexandre}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {4692--4701}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/janati20a/janati20a.pdf}, url = {http://proceedings.mlr.press/v119/janati20a.html}, abstract = {Entropy regularization in optimal transport (OT) has been the driver of many recent interests for Wasserstein metrics and barycenters in machine learning. It allows to keep the appealing geometrical properties of the unregularized Wasserstein distance while having a significantly lower complexity thanks to Sinkhorn’s algorithm. However, entropy brings some inherent smoothing bias, resulting for example in blurred barycenters. This side effect has prompted an increasing temptation in the community to settle for a slower algorithm such as log-domain stabilized Sinkhorn which breaks the parallel structure that can be leveraged on GPUs, or even go back to unregularized OT. Here we show how this bias is tightly linked to the reference measure that defines the entropy regularizer and propose debiased Sinkhorn barycenters that preserve the best of worlds: fast Sinkhorn-like iterations without entropy smoothing. Theoretically, we prove that this debiasing is perfect for Gaussian distributions with equal variance. Empirically, we illustrate the reduced blurring and the computational advantage.} }
Endnote
%0 Conference Paper %T Debiased Sinkhorn barycenters %A Hicham Janati %A Marco Cuturi %A Alexandre Gramfort %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-janati20a %I PMLR %P 4692--4701 %U http://proceedings.mlr.press/v119/janati20a.html %V 119 %X Entropy regularization in optimal transport (OT) has been the driver of many recent interests for Wasserstein metrics and barycenters in machine learning. It allows to keep the appealing geometrical properties of the unregularized Wasserstein distance while having a significantly lower complexity thanks to Sinkhorn’s algorithm. However, entropy brings some inherent smoothing bias, resulting for example in blurred barycenters. This side effect has prompted an increasing temptation in the community to settle for a slower algorithm such as log-domain stabilized Sinkhorn which breaks the parallel structure that can be leveraged on GPUs, or even go back to unregularized OT. Here we show how this bias is tightly linked to the reference measure that defines the entropy regularizer and propose debiased Sinkhorn barycenters that preserve the best of worlds: fast Sinkhorn-like iterations without entropy smoothing. Theoretically, we prove that this debiasing is perfect for Gaussian distributions with equal variance. Empirically, we illustrate the reduced blurring and the computational advantage.
APA
Janati, H., Cuturi, M. & Gramfort, A.. (2020). Debiased Sinkhorn barycenters. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:4692-4701 Available from http://proceedings.mlr.press/v119/janati20a.html.

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