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.
|Original language||American English|
|State||Published - 2020|
|Event||22nd International Conference on Artificial Intelligence and Statistics, AISTATS 2019 - Naha, Japan|
Duration: 16 Apr 2019 → 18 Apr 2019
|Conference||22nd International Conference on Artificial Intelligence and Statistics, AISTATS 2019|
|Period||16/04/19 → 18/04/19|
Bibliographical noteFunding Information:
M.N. is supported by the James S. McDonnell Foundation, Schmidt Futures, Israel Council for Higher Education, and the John Harvard Distinguished Science Fellows Program; P.R. by NSF grants DMS-1712596 and TRIPODS-1740751 and IIS-1838071, ONR grant N00014-17-1-2147, the Chan Zuckerberg Initiative DAF 2018-182642, and the MIT Skoltech Seed Fund; G.S. by a Burroughs Welcome Fund Career Award at the Scientific Interface and the Klarman Cell Observatory; and J.W. by the Josephine de Karman fellowship.
© 2019 by the author(s).