# Semi-dual Regularized Optimal Transport

arXiv: https://arxiv.org/abs/1811.05527 · HTML: https://mlsanity.com/paper/1811.05527/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Nov 2018.

## Built on by (most cited first, 26 of 26)

- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Flow Matching with Semidiscrete Couplings](https://mlsanity.com/paper/2509.25519/) (2025)
- [Energy-Guided Continuous Entropic Barycenter Estimation for General Costs](https://mlsanity.com/paper/2310.01105/) (2023)
- [Doubly Regularized Entropic Wasserstein Barycenters](https://mlsanity.com/paper/2303.11844/) (2023)
- [Variational Wasserstein Barycenters with c-Cyclical Monotonicity](https://mlsanity.com/paper/2110.11707/) (2021)
- [Importance Sparsification for Sinkhorn Algorithm](https://mlsanity.com/paper/2306.06581/) (2023)
- [A Stochastic Bregman Primal-Dual Splitting Algorithm for Composite Optimization](https://mlsanity.com/paper/2112.11928/) (2021)
- [Fast Dual Subgradient Optimization of the Integrated Transportation Distance Between Stochastic Kernels](https://mlsanity.com/paper/2312.01432/) (2023)
- [Randomized Stochastic Gradient Descent Ascent](https://mlsanity.com/paper/2111.13162/) (2021)
- [Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters](https://mlsanity.com/paper/2203.12136/) (2022)
- [A unified framework for non-negative matrix and tensor factorisations with a smoothed Wasserstein loss](https://mlsanity.com/paper/2104.01708/) (2021)
- [Averaging Spatio-temporal Signals using Optimal Transport and Soft Alignments](https://mlsanity.com/paper/2203.05813/) (2022)
- [Fully Probabilistic Design for Optimal Transport](https://mlsanity.com/paper/2212.13912/) (2022)
- [Faster Computation of Entropic Optimal Transport via Stable Low Frequency Modes](https://mlsanity.com/paper/2506.14780/) (2025)
- [Federated Calculation of the Free-Support Transportation Barycenter by Single-Loop Dual Decomposition](https://mlsanity.com/paper/2507.19627/) (2025)
- [Entropic Riemannian Neural Optimal Transport](https://mlsanity.com/paper/2605.04255/) (2026)
- [Estimation of Stationary Optimal Transport Plans](https://mlsanity.com/paper/2107.11858/) (2021)
- [Removing supervision in semantic segmentation with local-global matching and area balancing](https://mlsanity.com/paper/2303.17410/) (2023)
- [Computational Guarantees for Doubly Entropic Wasserstein Barycenters via Damped Sinkhorn Iterations](https://mlsanity.com/paper/2307.13370/) (2023)
- [Semi-Discrete Optimal Transport: Nearly Minimax Estimation With Stochastic Gradient Descent and Adaptive Entropic Regularization](https://mlsanity.com/paper/2405.14459/) (2024)
- [Designing Algorithms for Entropic Optimal Transport from an Optimisation Perspective](https://mlsanity.com/paper/2507.12246/) (2025)
- [Decreasing Entropic Regularization Averaged Gradient for Semi-Discrete Optimal Transport](https://mlsanity.com/paper/2510.27340/) (2025)
- [Stability of the Monge Map in Semi-Dual Optimal Transport](https://mlsanity.com/paper/2605.05569/) (2026)
- [Generative Modeling on Metric Graphs via Neural Optimal Transport](https://mlsanity.com/paper/2606.16273/) (2026)
- [Tight Nonasymptotic Local Convergence of Sinkhorn-Knopp](https://mlsanity.com/paper/2608.11760/) (2026)

## Abstract (arXiv)

Variational problems that involve Wasserstein distances and more generally optimal transport (OT) theory are playing an increasingly important role in data sciences. Such problems can be used to form an examplar measure out of various probability measures, as in the Wasserstein barycenter problem, or to carry out parametric inference and density fitting, where the loss is measured in terms of an optimal transport cost to the measure of observations. Despite being conceptually simple, such problems are computationally challenging because they involve minimizing over quantities (Wasserstein distances) that are themselves hard to compute. Entropic regularization has recently emerged as an efficient tool to approximate the solution of such variational Wasserstein problems. In this paper, we give a thorough duality tour of these regularization techniques. In particular, we show how important concepts from classical OT such as c-transforms and semi-discrete approaches translate into similar ideas in a regularized setting. These dual formulations lead to smooth variational problems, which can be solved using smooth, differentiable and convex optimization problems that are simpler to implement and numerically more stable that their un-regularized counterparts. We illustrate the versatility of this approach by applying it to the computation of Wasserstein barycenters and gradient flows of spatial regularization functionals.