# Optimal Transportation by Orthogonal Coupling Dynamics

arXiv: https://arxiv.org/abs/2410.08060 · HTML: https://mlsanity.com/paper/2410.08060/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 77 references that are in the map.
- First posted on arXiv 10 Oct 2024.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifold](https://mlsanity.com/paper/2408.14608/) (2024)
- [On the representation and learning of monotone triangular transport maps](https://mlsanity.com/paper/2009.10303/) (2020)
- [Structured Optimal Transport](https://mlsanity.com/paper/1712.06199/) (2017)
- [Dynamical Measure Transport and Neural PDE Solvers for Sampling](https://mlsanity.com/paper/2407.07873/) (2024)
- [Projected Langevin dynamics and a gradient flow for entropic optimal transport](https://mlsanity.com/paper/2309.08598/) (2023)
- [Stochastic Optimization for Regularized Wasserstein Estimators](https://mlsanity.com/paper/2002.08695/) (2020)
- [The linear conditional expectation in Hilbert space](https://mlsanity.com/paper/2008.12070/) (2020)
- [Moment-SoS Methods for Optimal Transport Problems](https://mlsanity.com/paper/2211.10742/) (2022)
- [Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processes](https://mlsanity.com/paper/2404.09402/) (2024)
- [Macroscopic Noisy Bounded Confidence Models with Distributed Radical Opinions](https://mlsanity.com/paper/1905.04057/) (2019)

## Built on by (most cited first, 2 of 2)

- [Collision-based Dynamics for Multi-Marginal Optimal Transport](https://mlsanity.com/paper/2412.16385/) (2024)
- [Bridging Reinforcement Learning and Optimal Control via Feasible Action Mapping](https://mlsanity.com/paper/2607.23930/) (2026)

## Abstract (arXiv)

Many numerical and learning algorithms rely on the solution of the Monge-Kantorovich problem and Wasserstein distances, which provide appropriate distributional metrics. While the natural approach is to treat the problem as an infinite-dimensional linear programming, such a methodology limits the computational performance due to the polynomial scaling with respect to the sample size along with intensive memory requirements. We propose a novel alternative framework to address the Monge-Kantorovich problem based on a projection type gradient descent scheme. The dynamics builds on the notion of the conditional expectation, where the connection with the opinion dynamics is leveraged to devise efficient numerical schemes. We demonstrate that the resulting dynamics recovers random maps with favourable computational performance. Along with the theoretical insight, the proposed dynamics paves the way for innovative approaches to construct numerical schemes for computing optimal transport maps as well as Wasserstein distances.