Permutation invariant networks to learn Wasserstein metrics

TL;DR

This paper introduces permutation invariant neural networks that learn to approximate Wasserstein distances between probability measures, enabling efficient comparison and moment estimation of distributions.

Contribution

The authors develop a neural network architecture that encodes probability measures into a space where Euclidean distances approximate Wasserstein metrics, generalizing to unseen data.

Findings

Network accurately estimates Wasserstein distances between distributions.

It generalizes well to unseen probability densities.

The model successfully learns first and second moments of distributions.

Abstract

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.