Statistical Optimal Transport via Factored Couplings
This work addresses the curse of dimensionality in data-driven optimal transport, with applications in fields like computational biology, though it appears incremental as it builds on existing optimal transport methods.
The authors tackled the problem of estimating Wasserstein distances and optimal transport plans from high-dimensional samples by introducing a new structural assumption called low transport rank, which led to significant improvements in tasks like domain adaptation for single-cell RNA sequencing data.
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.