Statistical Optimal Transport via Factored Couplings
Aden Forrow, Jan-Christian Hütter, Mor Nitzan, Philippe Rigollet, Geoffrey Schiebinger, Jonathan Weed
Abstract
We propose a new method to estimate Wasserstein distances and optimal transport plans between two probability distributions from samples in high dimension. Unlike plug-in rules that simply replace the true distributions by their empirical counterparts, our method promotes couplings with low transport rank, a new structural assumption that is similar to the nonnegative rank of a matrix. Regularizing based on this assumption leads to drastic improvements on high-dimensional data for various tasks, including domain adaptation in single-cell RNA sequencing data. These findings are supported by a theoretical analysis that indicates that the transport rank is key in overcoming the curse of dimensionality inherent to data-driven optimal transport.
BibTeX
@InProceedings{pmlr-v89-forrow19a,
title = {Statistical Optimal Transport via Factored Couplings},
author = {Forrow, Aden and H\"{u}tter, Jan-Christian and Nitzan, Mor and Rigollet, Philippe and Schiebinger, Geoffrey and Weed, Jonathan},
booktitle = {Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics},
pages = {2454--2465},
year = {2019},
editor = {Chaudhuri, Kamalika and Sugiyama, Masashi},
volume = {89},
series = {Proceedings of Machine Learning Research},
month = {16--18 Apr},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v89/forrow19a/forrow19a.pdf},
url = {https://proceedings.mlr.press/v89/forrow19a.html},
abstract = {We propose a new method to estimate Wasserstein distances and optimal transport plans between two probability distributions from samples in high dimension. Unlike plug-in rules that simply replace the true distributions by their empirical counterparts, our method promotes couplings with low transport rank, a new structural assumption that is similar to the nonnegative rank of a matrix. Regularizing based on this assumption leads to drastic improvements on high-dimensional data for various tasks, including domain adaptation in single-cell RNA sequencing data. These findings are supported by a theoretical analysis that indicates that the transport rank is key in overcoming the curse of dimensionality inherent to data-driven optimal transport.}
}