# A Smoothed Dual Approach for Variational Wasserstein Problems

arXiv: https://arxiv.org/abs/1503.02533 · HTML: https://mlsanity.com/paper/1503.02533/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Mar 2015.

## Built on by (most cited first, 29 of 29)

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- [Variational Wasserstein Barycenters with c-Cyclical Monotonicity](https://mlsanity.com/paper/2110.11707/) (2021)
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- [On Partial Optimal Transport: Revising the Infeasibility of Sinkhorn and Efficient Gradient Methods](https://mlsanity.com/paper/2312.13970/) (2023)
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- [Deep Unsupervised Domain Adaptation for Time Series Classification: a Benchmark](https://mlsanity.com/paper/2312.09857/) (2023)
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- [Continual Learning of Generative Models with Limited Data: From Wasserstein-1 Barycenter to Adaptive Coalescence](https://mlsanity.com/paper/2101.09225/) (2021)
- [A unified framework for non-negative matrix and tensor factorisations with a smoothed Wasserstein loss](https://mlsanity.com/paper/2104.01708/) (2021)
- [Algorithms for Weak Optimal Transport with an Application to Economics](https://mlsanity.com/paper/2205.09825/) (2022)
- [Complexity of Block Coordinate Descent with Proximal Regularization and Applications to Wasserstein CP-dictionary Learning](https://mlsanity.com/paper/2306.02420/) (2023)
- [Schr"odinger Bridge Matching for Tree-Structured Costs and Entropic Wasserstein Barycentres](https://mlsanity.com/paper/2506.17197/) (2025)
- [The GenCol algorithm for high-dimensional optimal transport: general formulation and application to barycenters and Wasserstein splines](https://mlsanity.com/paper/2209.09081/) (2022)
- [Relax and Merge: A Simple Yet Effective Framework for Solving Fair k-Means and k-sparse Wasserstein Barycenter Problems](https://mlsanity.com/paper/2411.01115/) (2024)
- [Federated Calculation of the Free-Support Transportation Barycenter by Single-Loop Dual Decomposition](https://mlsanity.com/paper/2507.19627/) (2025)
- [Mitigating Negative Transfer via Reducing Environmental Disagreement](https://mlsanity.com/paper/2510.24044/) (2025)
- [A Distributed Algorithm for Measure-valued Optimization with Additive Objective](https://mlsanity.com/paper/2202.08930/) (2022)
- [Beckman Defense](https://mlsanity.com/paper/2301.01495/) (2023)
- [Learning Concave Bid Shading Strategies in Online Auctions via Measure-valued Proximal Optimization](https://mlsanity.com/paper/2509.10693/) (2025)
- [Losses that Cook: Topological Optimal Transport for Structured Recipe Generation](https://mlsanity.com/paper/2601.02531/) (2026)
- [A Sub-4 Approximation for Fair k-Means](https://mlsanity.com/paper/2609.07974/) (2026)

## Abstract (arXiv)

Variational problems that involve Wasserstein distances have been recently proposed to summarize and learn from probability measures. Despite being conceptually simple, such problems are computationally challenging because they involve minimizing over quantities (Wasserstein distances) that are themselves hard to compute. We show that the dual formulation of Wasserstein variational problems introduced recently by Carlier et al. (2014) can be regularized using an entropic smoothing, which leads to smooth, differentiable, convex optimization problems that are simpler to implement and numerically more stable. We illustrate the versatility of this approach by applying it to the computation of Wasserstein barycenters and gradient flows of spacial regularization functionals.