# Regularized Optimal Transport and the Rot Mover's Distance

arXiv: https://arxiv.org/abs/1610.06447 · HTML: https://mlsanity.com/paper/1610.06447/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Oct 2016.

## Built on by (most cited first, 24 of 24)

- [Low-Rank Sinkhorn Factorization](https://mlsanity.com/paper/2103.04737/) (2021)
- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [Sparsity-Constrained Optimal Transport](https://mlsanity.com/paper/2209.15466/) (2022)
- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps](https://mlsanity.com/paper/2302.04065/) (2023)
- [Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution](https://mlsanity.com/paper/2103.06263/) (2021)
- [Learning High Dimensional Wasserstein Geodesics](https://mlsanity.com/paper/2102.02992/) (2021)
- [Score-Based Diffusion Policy Compatible with Reinforcement Learning via Optimal Transport](https://mlsanity.com/paper/2502.12631/) (2025)
- [A novel notion of barycenter for probability distributions based on optimal weak mass transport](https://mlsanity.com/paper/2102.13380/) (2021)
- [On the Existence of Optimal Transport Gradient for Learning Generative Models](https://mlsanity.com/paper/2102.05542/) (2021)
- [Optimal transport with f-divergence regularization and generalized Sinkhorn algorithm](https://mlsanity.com/paper/2105.14337/) (2021)
- [Revisiting Counterfactual Regression through the Lens of Gromov-Wasserstein Information Bottleneck](https://mlsanity.com/paper/2405.15505/) (2024)
- [Discrete Optimal Transport with Independent Marginals is #P-Hard](https://mlsanity.com/paper/2203.01161/) (2022)
- [Beyond kNN: Adaptive, Sparse Neighborhood Graphs via Optimal Transport](https://mlsanity.com/paper/2208.00604/) (2022)
- [Optimal Transport with Tempered Exponential Measures](https://mlsanity.com/paper/2309.04015/) (2023)
- [A Riemannian Approach to Ground Metric Learning for Optimal Transport](https://mlsanity.com/paper/2409.10085/) (2024)
- [SIDDA: SInkhorn Dynamic Domain Adaptation for Image Classification with Equivariant Neural Networks](https://mlsanity.com/paper/2501.14048/) (2025)
- [Relaxation of optimal transport problem via strictly convex functions](https://mlsanity.com/paper/2102.07336/) (2021)
- [Fully Probabilistic Design for Optimal Transport](https://mlsanity.com/paper/2212.13912/) (2022)
- [Keypoint-Guided Optimal Transport: Models, Algorithms, and Applications](https://mlsanity.com/paper/2303.13102/) (2023)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Convergence of Batch Greenkhorn for Regularized Multimarginal Optimal Transport](https://mlsanity.com/paper/2112.00838/) (2021)
- [Connecting adversarial attacks and optimal transport for domain adaptation](https://mlsanity.com/paper/2205.15424/) (2022)
- [Robust computation of optimal transport by β-potential regularization](https://mlsanity.com/paper/2212.13251/) (2022)

## Abstract (arXiv)

This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to Bregman divergences. Our framework thus naturally generalizes a previously proposed regularization based on the Boltzmann-Shannon entropy related to the Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We call the regularized optimal transport distance the rot mover's distance in reference to the classical earth mover's distance. We develop two generic schemes that we respectively call the alternate scaling algorithm and the non-negative alternate scaling algorithm, to compute efficiently the regularized optimal plans depending on whether the domain of the regularizer lies within the non-negative orthant or not. These schemes are based on Dykstra's algorithm with alternate Bregman projections, and further exploit the Newton-Raphson method when applied to separable divergences. We enhance the separable case with a sparse extension to deal with high data dimensions. We also instantiate our proposed framework and discuss the inherent specificities for well-known regularizers and statistical divergences in the machine learning and information geometry communities. Finally, we demonstrate the merits of our methods with experiments using synthetic data to illustrate the effect of different regularizers and penalties on the solutions, as well as real-world data for a pattern recognition application to audio scene classification.