  Understanding the space of probability measures on a metric space equipped
with a Wasserstein distance is one of the fundamental questions in mathematical
analysis. The Wasserstein metric has received a lot of attention in the machine
learning community especially for its principled way of comparing
distributions. In this work, we use a permutation invariant network to map
samples from probability measures into a low-dimensional space such that the
Euclidean distance between the encoded samples reflects the Wasserstein
distance between probability measures. We show that our network can generalize
to correctly compute distances between unseen densities. We also show that
these networks can learn the first and the second moments of probability
distributions.
