  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.
