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Sliced Multi-Marginal Optimal Transport
Samuel Cohen · Alexander Terenin · Yannik Pitcan · Brandon Amos · Marc Deisenroth · Senanayak Sesh Kumar Karri
Event URL: https://arxiv.org/abs/2102.07115 »

Multi-marginal optimal transport enables one to compare multiple probability measures, which increasingly finds application in multi-task learning problems.One practical limitation of multi-marginal transport is computational scalability in the number of measures, samples and dimensionality.In this work, we propose a multi-marginal optimal transport paradigm based on random one-dimensional projections, whose (generalized) distance we term the \emph{sliced multi-marginal Wasserstein distance}.To construct this distance, we introduce a characterization of the one-dimensional multi-marginal Kantorovich problem and use it to highlight a number of properties of the sliced multi-marginal Wasserstein distance. In particular, we show that (i) the sliced multi-marginal Wasserstein distance is a (generalized) metric that induces the same topology as the standard Wasserstein distance, (ii) it admits a dimension-free sample complexity, (iii) it is tightly connected with the problem of barycentric averaging under the sliced-Wasserstein metric.We conclude by illustrating the sliced multi-marginal Wasserstein on multi-task density estimation and multi-dynamics reinforcement learning problems.

Author Information

Samuel Cohen (University College London)
Alexander Terenin (University of Cambridge)
Yannik Pitcan (UC Berkeley)
Brandon Amos (Carnegie Mellon University)
Marc Deisenroth (University College London)
Marc Deisenroth

Professor Marc Deisenroth is the DeepMind Chair in Artificial Intelligence at University College London and the Deputy Director of UCL's Centre for Artificial Intelligence. He also holds a visiting faculty position at the University of Johannesburg and Imperial College London. Marc's research interests center around data-efficient machine learning, probabilistic modeling and autonomous decision making. Marc was Program Chair of EWRL 2012, Workshops Chair of RSS 2013, EXPO-Co-Chair of ICML 2020, and Tutorials Co-Chair of NeurIPS 2021. In 2019, Marc co-organized the Machine Learning Summer School in London. He received Paper Awards at ICRA 2014, ICCAS 2016, and ICML 2020. He is co-author of the book [Mathematics for Machine Learning](https://mml-book.github.io) published by Cambridge University Press (2020).

Senanayak Sesh Kumar Karri (Imperial College London)

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