Cohen, Samuel;
Terenin, Alexander;
Pitcan, Yannik;
Amos, Brandon;
Deisenroth, Marc Peter;
Kumar, KS Sesh;
(2021)
Sliced Multi-Marginal Optimal Transport.
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Abstract
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 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.
| Type: | Working / discussion paper |
|---|---|
| Title: | Sliced Multi-Marginal Optimal Transport |
| Event: | NeurIPS 2021: Thirty-fifth Conference on Neural Information Processing Systems |
| Open access status: | An open access version is available from UCL Discovery |
| Publisher version: | https://arxiv.org/abs/2102.07115v2 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher's terms and conditions. |
| UCL classification: | UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science UCL > Provost and Vice Provost Offices > UCL BEAMS UCL |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10142929 |
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