Statistical Optimal Transport via Factored Couplings

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 distri-butions by their empirical counterparts, our method promotes couplings with low trans-port rank, a new structural assumption that is similar to the nonnegative rank of a ma-trix. Regularizing based on this assump-tion leads to drastic improvements on high-dimensional data for various tasks, includ-ing domain adaptation in single-cell RNA sequencing data. These findings are sup-ported by a theoretical analysis that indicates that the transport rank is key in overcoming the curse of dimensionality inherent to data-driven optimal transport.