Efficient estimates of optimal transport via low-dimensional embeddings

TL;DR

This paper introduces a scalable method for approximating optimal transport distances in high-dimensional data by leveraging low-dimensional embeddings and neural networks, improving computational efficiency.

Contribution

It extends previous low-rank projection methods by using general 1-Lipschitz maps, including neural networks, to approximate OT distances efficiently.

Findings

Method scales well with high-dimensional data

Neural networks effectively approximate 1-Lipschitz maps

Achieves accurate OT estimates with reduced computational cost

Abstract

Optimal transport distances (OT) have been widely used in recent work in Machine Learning as ways to compare probability distributions. These are costly to compute when the data lives in high dimension. Recent work by Paty et al., 2019, aims specifically at reducing this cost by computing OT using low-rank projections of the data (seen as discrete measures). We extend this approach and show that one can approximate OT distances by using more general families of maps provided they are 1-Lipschitz. The best estimate is obtained by maximising OT over the given family. As OT calculations are done after mapping data to a lower dimensional space, our method scales well with the original data dimension. We demonstrate the idea with neural networks.