  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.
