Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics

Tam Le, Truyen Nguyen, Dinh Phung, Viet Anh Nguyen
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:9844-9868, 2022.

Abstract

Optimal transport (OT) is a popular measure to compare probability distributions. However, OT suffers a few drawbacks such as (i) a high complexity for computation, (ii) indefiniteness which limits its applicability to kernel machines. In this work, we consider probability measures supported on a graph metric space and propose a novel Sobolev transport metric. We show that the Sobolev transport metric yields a closed-form formula for fast computation and it is negative definite. We show that the space of probability measures endowed with this transport distance is isometric to a bounded convex set in a Euclidean space with a weighted l_p distance. We further exploit the negative definiteness of the Sobolev transport to design positive-definite kernels, and evaluate their performances against other baselines in document classification with word embeddings and in topological data analysis.

Cite this Paper

BibTeX
@InProceedings{pmlr-v151-le22b, title = { Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics }, author = {Le, Tam and Nguyen, Truyen and Phung, Dinh and Anh Nguyen, Viet}, booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics}, pages = {9844--9868}, year = {2022}, editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel}, volume = {151}, series = {Proceedings of Machine Learning Research}, month = {28--30 Mar}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v151/le22b/le22b.pdf}, url = {https://proceedings.mlr.press/v151/le22b.html}, abstract = { Optimal transport (OT) is a popular measure to compare probability distributions. However, OT suffers a few drawbacks such as (i) a high complexity for computation, (ii) indefiniteness which limits its applicability to kernel machines. In this work, we consider probability measures supported on a graph metric space and propose a novel Sobolev transport metric. We show that the Sobolev transport metric yields a closed-form formula for fast computation and it is negative definite. We show that the space of probability measures endowed with this transport distance is isometric to a bounded convex set in a Euclidean space with a weighted l_p distance. We further exploit the negative definiteness of the Sobolev transport to design positive-definite kernels, and evaluate their performances against other baselines in document classification with word embeddings and in topological data analysis. } }
Endnote
%0 Conference Paper %T Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics %A Tam Le %A Truyen Nguyen %A Dinh Phung %A Viet Anh Nguyen %B Proceedings of The 25th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2022 %E Gustau Camps-Valls %E Francisco J. R. Ruiz %E Isabel Valera %F pmlr-v151-le22b %I PMLR %P 9844--9868 %U https://proceedings.mlr.press/v151/le22b.html %V 151 %X Optimal transport (OT) is a popular measure to compare probability distributions. However, OT suffers a few drawbacks such as (i) a high complexity for computation, (ii) indefiniteness which limits its applicability to kernel machines. In this work, we consider probability measures supported on a graph metric space and propose a novel Sobolev transport metric. We show that the Sobolev transport metric yields a closed-form formula for fast computation and it is negative definite. We show that the space of probability measures endowed with this transport distance is isometric to a bounded convex set in a Euclidean space with a weighted l_p distance. We further exploit the negative definiteness of the Sobolev transport to design positive-definite kernels, and evaluate their performances against other baselines in document classification with word embeddings and in topological data analysis.
APA
Le, T., Nguyen, T., Phung, D. & Anh Nguyen, V.. (2022). Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics . Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 151:9844-9868 Available from https://proceedings.mlr.press/v151/le22b.html.

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