Efficient estimates of optimal transport via low-dimensional embeddings

Patric Fulop, Vincent Danos

Keywords: optimal transport, sinkhorn divergences, robustness, neural networks, lipschitz, spectral norm

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 aims specifically at reducing this cost by computing OT using low-rank projections of the data (seen as discrete measures)~\citep{paty2019subspace}. 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.

One-sentence Summary: Approximating optimal transport distances using a low-dimensional space constructed through a general projection map that is 1-Lipschitz.

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics

Reviewed Version (pdf): https://openreview.net/references/pdf?id=AVF2_Ia5I0