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Understanding the space of probability measures on a metric space equippedwith a Wasserstein distance is one of the fundamental questions in mathematicalanalysis. The Wasserstein metric has received a lot of attention in the machinelearning community especially for its principled way of comparingdistributions. In this work, we use a permutation invariant network to mapsamples from probability measures into a low-dimensional space such that theEuclidean distance between the encoded samples reflects the Wassersteindistance between probability measures. We show that our network can generalizeto correctly compute distances between unseen densities. We also show thatthese networks can learn the first and the second moments of probabilitydistributions.

Permutation invariant networks to learn Wasserstein metrics