Sliced Multi-Marginal Optimal Transport

• URL: http://arxiv.org/abs/2102.07115v1
• Date: Sun, 14 Feb 2021 09:58:47 GMT
• Title: Sliced Multi-Marginal Optimal Transport
• Authors: Samuel Cohen, K S Sesh Kumar, Marc Peter Deisenroth
• Abstract summary: We study multi-marginal optimal transport, a generalization of optimal transport that allows us to define discrepancies between multiple measures. We show that computing the sliced multi-marginal discrepancy is massively scalable for a large number of probability measures with support as large as $107$ samples.
• Score: 21.82052188474956
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study multi-marginal optimal transport, a generalization of optimal transport that allows us to define discrepancies between multiple measures. It provides a framework to solve multi-task learning problems and to perform barycentric averaging. However, multi-marginal distances between multiple measures are typically challenging to compute because they require estimating a transport plan with $N^P$ variables. In this paper, we address this issue in the following way: 1) we efficiently solve the one-dimensional multi-marginal Monge-Wasserstein problem for a classical cost function in closed form, and 2) we propose a higher-dimensional multi-marginal discrepancy via slicing and study its generalized metric properties. We show that computing the sliced multi-marginal discrepancy is massively scalable for a large number of probability measures with support as large as $10^7$ samples. Our approach can be applied to solving problems such as barycentric averaging, multi-task density estimation and multi-task reinforcement learning.

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