  The Wasserstein barycenter is a geometric construct which captures the notion
of centrality among probability distributions, and which has found many
applications in machine learning. However, most algorithms for finding even an
approximate barycenter suffer an exponential dependence on the dimension $d$ of
the underlying space of the distributions. In order to cope with this "curse of
dimensionality," we study dimensionality reduction techniques for the
Wasserstein barycenter problem. When the barycenter is restricted to support of
size $n$, we show that randomized dimensionality reduction can be used to map
the problem to a space of dimension $O(\log n)$ independent of both $d$ and
$k$, and that \emph{any} solution found in the reduced dimension will have its
cost preserved up to arbitrary small error in the original space. We provide
matching upper and lower bounds on the size of the reduced dimension, showing
that our methods are optimal up to constant factors. We also provide a coreset
construction for the Wasserstein barycenter problem that significantly
decreases the number of input distributions. The coresets can be used in
conjunction with random projections and thus further improve computation time.
Lastly, our experimental results validate the speedup provided by
dimensionality reduction while maintaining solution quality.
