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Tree-Sliced Variants of Wasserstein Distances
Tam Le · Makoto Yamada · Kenji Fukumizu · Marco Cuturi

Tue Dec 10 10:45 AM -- 12:45 PM (PST) @ East Exhibition Hall B + C #80

Optimal transport (\OT) theory defines a powerful set of tools to compare probability distributions. \OT~suffers however from a few drawbacks, computational and statistical, which have encouraged the proposal of several regularized variants of OT in the recent literature, one of the most notable being the \textit{sliced} formulation, which exploits the closed-form formula between univariate distributions by projecting high-dimensional measures onto random lines. We consider in this work a more general family of ground metrics, namely \textit{tree metrics}, which also yield fast closed-form computations and negative definite, and of which the sliced-Wasserstein distance is a particular case (the tree is a chain). We propose the tree-sliced Wasserstein distance, computed by averaging the Wasserstein distance between these measures using random tree metrics, built adaptively in either low or high-dimensional spaces. Exploiting the negative definiteness of that distance, we also propose a positive definite kernel, and test it against other baselines on a few benchmark tasks.

Author Information


I completed my Ph.D. in 09/2015, and officially obtained a Ph.D. degree from Kyoto University in 01/2016, under the supervision of Professor Marco Cuturi and Professor Akihiro Yamamoto. Then, I worked as a post-doctoral researcher at the Nagoya Institute of Technology and National Institute of Materials Science, Japan between 01/2016 and 08/2017. After that, I have been working as a postdoctoral researcher in RIKEN AIP, Japan since 09/2017.

Makoto Yamada (Kyoto University / RIKEN AIP)
Kenji Fukumizu (Institute of Statistical Mathematics / Preferred Networks / RIKEN AIP)
Marco Cuturi (Google Brain & CREST - ENSAE)

Marco Cuturi is a research scientist at Google AI, Brain team in Paris. He received his Ph.D. in 11/2005 from the Ecole des Mines de Paris in applied mathematics. Before that he graduated from National School of Statistics (ENSAE) with a master degree (MVA) from ENS Cachan. He worked as a post-doctoral researcher at the Institute of Statistical Mathematics, Tokyo, between 11/2005 and 3/2007 and then in the financial industry between 4/2007 and 9/2008. After working at the ORFE department of Princeton University as a lecturer between 2/2009 and 8/2010, he was at the Graduate School of Informatics of Kyoto University between 9/2010 and 9/2016 as a tenured associate professor. He joined ENSAE in 9/2016 as a professor, where he is now working part-time. His main employment is now with Google AI (Brain team in Paris) since 10/2018, as a research scientist working on fundamental aspects of machine learning.

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