  Motivated by the statistical and computational challenges of computing
Wasserstein distances in high-dimensional contexts, machine learning
researchers have defined modified Wasserstein distances based on computing
distances between one-dimensional projections of the measures. Different
choices of how to aggregate these projected distances (averaging, random
sampling, maximizing) give rise to different distances, requiring different
statistical analyses. We define the \emph{Sliced Wasserstein Process}, a
stochastic process defined by the empirical Wasserstein distance between
projections of empirical probability measures to all one-dimensional subspaces,
and prove a uniform distributional limit theorem for this process. As a result,
we obtain a unified framework in which to prove distributional limit results
for all Wasserstein distances based on one-dimensional projections. We
illustrate these results on a number of examples where no distributional limits
were previously known.
