# Low-rank Optimal Transport: Approximation, Statistics and Debiasing

arXiv: https://arxiv.org/abs/2205.12365 · HTML: https://mlsanity.com/paper/2205.12365/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 35 references that are in the map.
- First posted on arXiv 24 May 2022.

## Builds on (most cited first, 16 of 16)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [Low-Rank Sinkhorn Factorization](https://mlsanity.com/paper/2103.04737/) (2021)
- [Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs](https://mlsanity.com/paper/2106.01128/) (2021)
- [Statistical Optimal Transport via Factored Couplings](https://mlsanity.com/paper/1806.07348/) (2018)
- [Wasserstein Dictionary Learning: Optimal Transport-based unsupervised non-linear dictionary learning](https://mlsanity.com/paper/1708.01955/) (2017)
- [Stochastic Mirror Descent: Convergence Analysis and Adaptive Variants via the Mirror Stochastic Polyak Stepsize](https://mlsanity.com/paper/2110.15412/) (2021)
- [Ground Metric Learning on Graphs](https://mlsanity.com/paper/1911.03117/) (2019)
- [Linear Time Sinkhorn Divergences using Positive Features](https://mlsanity.com/paper/2006.07057/) (2020)
- [Mirror Descent and Convex Optimization Problems With Non-Smooth Inequality Constraints](https://mlsanity.com/paper/1710.06612/) (2017)
- [Entropic regularization of continuous optimal transport problems](https://mlsanity.com/paper/1906.01333/) (2019)
- [Multi-subject MEG/EEG source imaging with sparse multi-task regression](https://mlsanity.com/paper/1910.01914/) (2019)
- [Approximating Optimal Transport via Low-rank and Sparse Factorization](https://mlsanity.com/paper/2111.06546/) (2021)

## Built on by (most cited first, 21 of 21)

- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [PLANETALIGN: A Comprehensive Python Library for Benchmarking Network Alignment](https://mlsanity.com/paper/2505.21366/) (2025)
- [Hierarchical Refinement: Optimal Transport to Infinity and Beyond](https://mlsanity.com/paper/2503.03025/) (2025)
- [Linear Optimal Partial Transport Embedding](https://mlsanity.com/paper/2302.03232/) (2023)
- [Unbalanced Low-rank Optimal Transport Solvers](https://mlsanity.com/paper/2305.19727/) (2023)
- [Meta-Learning for Unsupervised Outlier Detection with Optimal Transport](https://mlsanity.com/paper/2211.00372/) (2022)
- [LOTFormer: Doubly-Stochastic Linear Attention via Low-Rank Optimal Transport](https://mlsanity.com/paper/2509.23436/) (2025)
- [Efficient Transferable Optimal Transport via Min-Sliced Transport Plans](https://mlsanity.com/paper/2511.19741/) (2025)
- [Sample Complexity Using Infinite Multiview Models](https://mlsanity.com/paper/2302.04292/) (2023)
- [Partial Transport for Point-Cloud Registration](https://mlsanity.com/paper/2309.15787/) (2023)
- [Amortized Optimal Transport from Sliced Potentials](https://mlsanity.com/paper/2604.15114/) (2026)
- [Sliced-Regularized Optimal Transport](https://mlsanity.com/paper/2604.23944/) (2026)
- [Graph Cuts with Arbitrary Size Constraints Through Optimal Transport](https://mlsanity.com/paper/2402.04732/) (2024)
- [Linear Partial Gromov-Wasserstein Embedding](https://mlsanity.com/paper/2410.16669/) (2024)
- [CLAMS: A System for Zero-Shot Model Selection for Clustering](https://mlsanity.com/paper/2407.11286/) (2024)
- [Automated Machine Learning for Unsupervised Tabular Tasks](https://mlsanity.com/paper/2510.07569/) (2025)
- [Transport Clustering: Solving Low-Rank Optimal Transport via Clustering](https://mlsanity.com/paper/2603.03578/) (2026)
- [Multiscale Supervised Unbalanced Optimal Transport Flow Matching](https://mlsanity.com/paper/2605.16529/) (2026)
- [A Biconvex Formulation for Stable Transport of Mixture Models with a Unique Solution](https://mlsanity.com/paper/2606.02515/) (2026)
- [Cluster-Aware Matching via Laplacian Optimal Transport](https://mlsanity.com/paper/2607.16178/) (2026)
- [SinkSLOT: Sinkhorn via Sparse Lifted Optimal Transport](https://mlsanity.com/paper/2608.28262/) (2026)

## Abstract (arXiv)

The matching principles behind optimal transport (OT) play an increasingly important role in machine learning, a trend which can be observed when OT is used to disambiguate datasets in applications (e.g. single-cell genomics) or used to improve more complex methods (e.g. balanced attention in transformers or self-supervised learning). To scale to more challenging problems, there is a growing consensus that OT requires solvers that can operate on millions, not thousands, of points. The low-rank optimal transport (LOT) approach advocated in citescetbon2021lowrank holds several promises in that regard, and was shown to complement more established entropic regularization approaches, being able to insert itself in more complex pipelines, such as quadratic OT. LOT restricts the search for low-cost couplings to those that have a low-nonnegative rank, yielding linear time algorithms in cases of interest. However, these promises can only be fulfilled if the LOT approach is seen as a legitimate contender to entropic regularization when compared on properties of interest, where the scorecard typically includes theoretical properties (statistical complexity and relation to other methods) or practical aspects (debiasing, hyperparameter tuning, initialization). We target each of these areas in this paper in order to cement the impact of low-rank approaches in computational OT.