Fast and Smooth Interpolation on Wasserstein Space

Sinho Chewi, Julien Clancy, Thibaut Le Gouic, Philippe Rigollet, George Stepaniants, Austin Stromme
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:3061-3069, 2021.

Abstract

We propose a new method for smoothly interpolating probability measures using the geometry of optimal transport. To that end, we reduce this problem to the classical Euclidean setting, allowing us to directly leverage the extensive toolbox of spline interpolation. Unlike previous approaches to measure-valued splines, our interpolated curves (i) have a clear interpretation as governing particle flows, which is natural for applications, and (ii) come with the first approximation guarantees on Wasserstein space. Finally, we demonstrate the broad applicability of our interpolation methodology by fitting surfaces of measures using thin-plate splines.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-chewi21a, title = { Fast and Smooth Interpolation on Wasserstein Space }, author = {Chewi, Sinho and Clancy, Julien and Le Gouic, Thibaut and Rigollet, Philippe and Stepaniants, George and Stromme, Austin}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {3061--3069}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/chewi21a/chewi21a.pdf}, url = {https://proceedings.mlr.press/v130/chewi21a.html}, abstract = { We propose a new method for smoothly interpolating probability measures using the geometry of optimal transport. To that end, we reduce this problem to the classical Euclidean setting, allowing us to directly leverage the extensive toolbox of spline interpolation. Unlike previous approaches to measure-valued splines, our interpolated curves (i) have a clear interpretation as governing particle flows, which is natural for applications, and (ii) come with the first approximation guarantees on Wasserstein space. Finally, we demonstrate the broad applicability of our interpolation methodology by fitting surfaces of measures using thin-plate splines. } }
Endnote
%0 Conference Paper %T Fast and Smooth Interpolation on Wasserstein Space %A Sinho Chewi %A Julien Clancy %A Thibaut Le Gouic %A Philippe Rigollet %A George Stepaniants %A Austin Stromme %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-chewi21a %I PMLR %P 3061--3069 %U https://proceedings.mlr.press/v130/chewi21a.html %V 130 %X We propose a new method for smoothly interpolating probability measures using the geometry of optimal transport. To that end, we reduce this problem to the classical Euclidean setting, allowing us to directly leverage the extensive toolbox of spline interpolation. Unlike previous approaches to measure-valued splines, our interpolated curves (i) have a clear interpretation as governing particle flows, which is natural for applications, and (ii) come with the first approximation guarantees on Wasserstein space. Finally, we demonstrate the broad applicability of our interpolation methodology by fitting surfaces of measures using thin-plate splines.
APA
Chewi, S., Clancy, J., Le Gouic, T., Rigollet, P., Stepaniants, G. & Stromme, A.. (2021). Fast and Smooth Interpolation on Wasserstein Space . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:3061-3069 Available from https://proceedings.mlr.press/v130/chewi21a.html.

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