  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.
