# Regularized Discrete Optimal Transport

arXiv: https://arxiv.org/abs/1307.5551 · HTML: https://mlsanity.com/paper/1307.5551/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2013.

## Built on by (most cited first, 30 of 54)

- [ScanNet++: A High-Fidelity Dataset of 3D Indoor Scenes](https://mlsanity.com/paper/2308.11417/) (2023)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Hierarchical Multi-Marginal Optimal Transport for Network Alignment](https://mlsanity.com/paper/2310.04470/) (2023)
- [Sliced Optimal Partial Transport](https://mlsanity.com/paper/2212.08049/) (2022)
- [Mapping conditional distributions for domain adaptation under generalized target shift](https://mlsanity.com/paper/2110.15057/) (2021)
- [CSOT: Curriculum and Structure-Aware Optimal Transport for Learning with Noisy Labels](https://mlsanity.com/paper/2312.06221/) (2023)
- [On Transportation of Mini-batches: A Hierarchical Approach](https://mlsanity.com/paper/2102.05912/) (2021)
- [Large-scale optimal transport map estimation using projection pursuit](https://mlsanity.com/paper/2106.05838/) (2021)
- [Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution](https://mlsanity.com/paper/2103.06263/) (2021)
- [Hierarchical Optimal Transport for Unsupervised Domain Adaptation](https://mlsanity.com/paper/2112.02073/) (2021)
- [Learning Canonical View Representation for 3D Shape Recognition with Arbitrary Views](https://mlsanity.com/paper/2108.07084/) (2021)
- [Entropy Regularized Optimal Transport Independence Criterion](https://mlsanity.com/paper/2112.15265/) (2021)
- [InfoOT: Information Maximizing Optimal Transport](https://mlsanity.com/paper/2210.03164/) (2022)
- [Manifold optimization for non-linear optimal transport problems](https://mlsanity.com/paper/2103.00902/) (2021)
- [Fill the Gap: Quantifying and Reducing the Modality Gap in Image-Text Representation Learning](https://mlsanity.com/paper/2505.03703/) (2025)
- [Unbalanced CO-Optimal Transport](https://mlsanity.com/paper/2205.14923/) (2022)
- [Importance Sparsification for Sinkhorn Algorithm](https://mlsanity.com/paper/2306.06581/) (2023)
- [Cycle Consistent Probability Divergences Across Different Spaces](https://mlsanity.com/paper/2111.11328/) (2021)
- [Block-coordinate Frank-Wolfe algorithm and convergence analysis for semi-relaxed optimal transport problem](https://mlsanity.com/paper/2205.13766/) (2022)
- [Context-aware attention layers coupled with optimal transport domain adaptation and multimodal fusion methods for recognizing dementia from spontaneous speech](https://mlsanity.com/paper/2305.16406/) (2023)
- [Fast Optimal Transport through Sliced Wasserstein Generalized Geodesics](https://mlsanity.com/paper/2307.01770/) (2023)
- [Building a Winning Team: Selecting Source Model Ensembles using a Submodular Transferability Estimation Approach](https://mlsanity.com/paper/2309.02429/) (2023)
- [PINS: Proximal Iterations with Sparse Newton and Sinkhorn for Optimal Transport](https://mlsanity.com/paper/2502.03749/) (2025)
- [Optimal Transport-Guided Source-Free Adaptation for Face Anti-Spoofing](https://mlsanity.com/paper/2503.22984/) (2025)
- [Optimal transport with f-divergence regularization and generalized Sinkhorn algorithm](https://mlsanity.com/paper/2105.14337/) (2021)
- [An optimal transport approach for selecting a representative subsample with application in efficient kernel density estimation](https://mlsanity.com/paper/2206.01182/) (2022)
- [Fast block-coordinate Frank-Wolfe algorithm for semi-relaxed optimal transport](https://mlsanity.com/paper/2103.05857/) (2021)

## Abstract (arXiv)

This article introduces a generalization of the discrete optimal transport, with applications to color image manipulations. This new formulation includes a relaxation of the mass conservation constraint and a regularization term. These two features are crucial for image processing tasks, which necessitate to take into account families of multimodal histograms, with large mass variation across modes. The corresponding relaxed and regularized transportation problem is the solution of a convex optimization problem. Depending on the regularization used, this minimization can be solved using standard linear programming methods or first order proximal splitting schemes. The resulting transportation plan can be used as a color transfer map, which is robust to mass variation across images color palettes. Furthermore, the regularization of the transport plan helps to remove colorization artifacts due to noise amplification. We also extend this framework to the computation of barycenters of distributions. The barycenter is the solution of an optimization problem, which is separately convex with respect to the barycenter and the transportation plans, but not jointly convex. A block coordinate descent scheme converges to a stationary point of the energy. We show that the resulting algorithm can be used for color normalization across several images. The relaxed and regularized barycenter defines a common color palette for those images. Applying color transfer toward this average palette performs a color normalization of the input images.