# Statistical Optimal Transport via Factored Couplings

arXiv: https://arxiv.org/abs/1806.07348 · HTML: https://mlsanity.com/paper/1806.07348/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jun 2018.

## Built on by (most cited first, 30 of 38)

- [Low-Rank Sinkhorn Factorization](https://mlsanity.com/paper/2103.04737/) (2021)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs](https://mlsanity.com/paper/2106.01128/) (2021)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Meta Optimal Transport](https://mlsanity.com/paper/2206.05262/) (2022)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Low-rank Optimal Transport: Approximation, Statistics and Debiasing](https://mlsanity.com/paper/2205.12365/) (2022)
- [Minimax estimation of discontinuous optimal transport maps: The semi-discrete case](https://mlsanity.com/paper/2301.11302/) (2023)
- [Martingale Methods for Sequential Estimation of Convex Functionals and Divergences](https://mlsanity.com/paper/2103.09267/) (2021)
- [A Pseudo-Metric between Probability Distributions based on Depth-Trimmed Regions](https://mlsanity.com/paper/2103.12711/) (2021)
- [Scalable Optimal Transport in High Dimensions for Graph Distances, Embedding Alignment, and More](https://mlsanity.com/paper/2107.06876/) (2021)
- [Hierarchical Refinement: Optimal Transport to Infinity and Beyond](https://mlsanity.com/paper/2503.03025/) (2025)
- [Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach](https://mlsanity.com/paper/2304.06549/) (2023)
- [Unbalanced Low-rank Optimal Transport Solvers](https://mlsanity.com/paper/2305.19727/) (2023)
- [Improving Approximate Optimal Transport Distances using Quantization](https://mlsanity.com/paper/2102.12731/) (2021)
- [Fast Optimal Transport through Sliced Wasserstein Generalized Geodesics](https://mlsanity.com/paper/2307.01770/) (2023)
- [Order Constraints in Optimal Transport](https://mlsanity.com/paper/2110.07275/) (2021)
- [Approximating Optimal Transport via Low-rank and Sparse Factorization](https://mlsanity.com/paper/2111.06546/) (2021)
- [Matching for causal effects via multimarginal unbalanced optimal transport](https://mlsanity.com/paper/2112.04398/) (2021)
- [Tangential Wasserstein Projections](https://mlsanity.com/paper/2207.14727/) (2022)
- [Gromov-Wasserstein-like Distances in the Gaussian Mixture Models Space](https://mlsanity.com/paper/2310.11256/) (2023)
- [Efficient approximation of Earth Mover's Distance Based on Nearest Neighbor Search](https://mlsanity.com/paper/2401.07378/) (2024)
- [Factored couplings in multi-marginal optimal transport via difference of convex programming](https://mlsanity.com/paper/2110.00629/) (2021)
- [Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization](https://mlsanity.com/paper/2502.07510/) (2025)
- [Spherical Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11249/) (2025)
- [Budget-Constrained Bounds for Mini-Batch Estimation of Optimal Transport](https://mlsanity.com/paper/2210.13630/) (2022)
- [SAVA: Scalable Learning-Agnostic Data Valuation](https://mlsanity.com/paper/2406.01130/) (2024)
- [Optimal Transport for Probabilistic Circuits](https://mlsanity.com/paper/2410.13061/) (2024)
- [Fair Clustering via Alignment](https://mlsanity.com/paper/2505.09131/) (2025)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)

## Abstract (arXiv)

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.