# Low-Rank Optimal Transport through Factor Relaxation with Latent Coupling

arXiv: https://arxiv.org/abs/2411.10555 · HTML: https://mlsanity.com/paper/2411.10555/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 57 references that are in the map.
- First posted on arXiv 15 Nov 2024.

## Builds on (most cited first, 3 of 3)

- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Approximating Optimal Transport via Low-rank and Sparse Factorization](https://mlsanity.com/paper/2111.06546/) (2021)

## Built on by (most cited first, 7 of 7)

- [Hierarchical Refinement: Optimal Transport to Infinity and Beyond](https://mlsanity.com/paper/2503.03025/) (2025)
- [A Riemannian Approach to Low-Rank Optimal Transport](https://mlsanity.com/paper/2606.12120/) (2026)
- [Simplifying Optimal Transport through Schatten-p Regularization](https://mlsanity.com/paper/2510.11910/) (2025)
- [Transport Clustering: Solving Low-Rank Optimal Transport via Clustering](https://mlsanity.com/paper/2603.03578/) (2026)
- [A Biconvex Formulation for Stable Transport of Mixture Models with a Unique Solution](https://mlsanity.com/paper/2606.02515/) (2026)
- [Exact Rank-Space KL Projection for Shared-Marginal Low-Rank Factors: Application to Doubly Stochastic Clustering](https://mlsanity.com/paper/2608.08642/) (2026)
- [Dual-guided Hierarchical Edge Localization for Large-scale Optimal Transport Across Dimensions](https://mlsanity.com/paper/2609.13010/) (2026)

## Abstract (arXiv)

Optimal transport (OT) is a general framework for finding a minimum-cost transport plan, or coupling, between probability distributions, and has many applications in machine learning. A key challenge in applying OT to massive datasets is the quadratic scaling of the coupling matrix with the size of the dataset. [Forrow et al. 2019] introduced a factored coupling for the k-Wasserstein barycenter problem, which [Scetbon et al. 2021] adapted to solve the primal low-rank OT problem. We derive an alternative parameterization of the low-rank problem based on the textitlatent coupling (LC) factorization previously introduced by [Lin et al. 2021] generalizing [Forrow et al. 2019]. The LC factorization has multiple advantages for low-rank OT including decoupling the problem into three OT problems and greater flexibility and interpretability. We leverage these advantages to derive a new algorithm textitFactor Relaxation with Latent Coupling (FRLC), which uses textitcoordinate mirror descent to compute the LC factorization. FRLC handles multiple OT objectives (Wasserstein, Gromov-Wasserstein, Fused Gromov-Wasserstein), and marginal constraints (balanced, unbalanced, and semi-relaxed) with linear space complexity. We provide theoretical results on FRLC, and demonstrate superior performance on diverse applications -- including graph clustering and spatial transcriptomics -- while demonstrating its interpretability.