# Sinkhorn Divergences for Unbalanced Optimal Transport

arXiv: https://arxiv.org/abs/1910.12958 · HTML: https://mlsanity.com/paper/1910.12958/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Oct 2019.

## Built on by (most cited first, 30 of 55)

- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [Unbalanced Optimal Transport, from Theory to Numerics](https://mlsanity.com/paper/2211.08775/) (2022)
- [DiffSkill: Skill Abstraction from Differentiable Physics for Deformable Object Manipulations with Tools](https://mlsanity.com/paper/2203.17275/) (2022)
- [Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport](https://mlsanity.com/paper/2305.14777/) (2023)
- [Accurate Point Cloud Registration with Robust Optimal Transport](https://mlsanity.com/paper/2111.00648/) (2021)
- [Optimal Transport for Treatment Effect Estimation](https://mlsanity.com/paper/2310.18286/) (2023)
- [Unbalanced Optimal Transport through Non-negative Penalized Linear Regression](https://mlsanity.com/paper/2106.04145/) (2021)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Addressing Misspecification in Simulation-based Inference through Data-driven Calibration](https://mlsanity.com/paper/2405.08719/) (2024)
- [DexDeform: Dexterous Deformable Object Manipulation with Human Demonstrations and Differentiable Physics](https://mlsanity.com/paper/2304.03223/) (2023)
- [Scalable Wasserstein Gradient Flow for Generative Modeling through Unbalanced Optimal Transport](https://mlsanity.com/paper/2402.05443/) (2024)
- [Improving Neural Cross-Lingual Summarization via Employing Optimal Transport Distance for Knowledge Distillation](https://mlsanity.com/paper/2112.03473/) (2021)
- [Faster Unbalanced Optimal Transport: Translation invariant Sinkhorn and 1-D Frank-Wolfe](https://mlsanity.com/paper/2201.00730/) (2022)
- [Direct Measure Matching for Crowd Counting](https://mlsanity.com/paper/2107.01558/) (2021)
- [Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2402.04613/) (2024)
- [Rethinking the Representation in Federated Unsupervised Learning with Non-IID Data](https://mlsanity.com/paper/2403.16398/) (2024)
- [Fast and Scalable Optimal Transport for Brain Tractograms](https://mlsanity.com/paper/2107.02010/) (2021)
- [Scalable Unbalanced Sobolev Transport for Measures on a Graph](https://mlsanity.com/paper/2302.12498/) (2023)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [Diffeomorphic Mesh Deformation via Efficient Optimal Transport for Cortical Surface Reconstruction](https://mlsanity.com/paper/2305.17555/) (2023)
- [Consistent Optimal Transport with Empirical Conditional Measures](https://mlsanity.com/paper/2305.15901/) (2023)
- [Learning Generalizable Tool-use Skills through Trajectory Generation](https://mlsanity.com/paper/2310.00156/) (2023)
- [Partial Wasserstein Adversarial Network for Non-rigid Point Set Registration](https://mlsanity.com/paper/2203.02227/) (2022)
- [Topological Node2vec: Enhanced Graph Embedding via Persistent Homology](https://mlsanity.com/paper/2309.08241/) (2023)
- [Uncertainty-Aware Knowledge Distillation for Compact and Efficient 6DoF Pose Estimation](https://mlsanity.com/paper/2503.13053/) (2025)
- [Matching for causal effects via multimarginal unbalanced optimal transport](https://mlsanity.com/paper/2112.04398/) (2021)
- [D-Cubed: Latent Diffusion Trajectory Optimisation for Dexterous Deformable Manipulation](https://mlsanity.com/paper/2403.12861/) (2024)
- [DynaMo: Accelerating Language Model Inference with Dynamic Multi-Token Sampling](https://mlsanity.com/paper/2405.00888/) (2024)
- [Inferring stochastic dynamics with growth from cross-sectional data](https://mlsanity.com/paper/2505.13197/) (2025)
- [Zero-Shot Offline Imitation Learning via Optimal Transport](https://mlsanity.com/paper/2410.08751/) (2024)

## Abstract (arXiv)

Optimal transport induces the Earth Mover's (Wasserstein) distance between probability distributions, a geometric divergence that is relevant to a wide range of problems. Over the last decade, two relaxations of optimal transport have been studied in depth: unbalanced transport, which is robust to the presence of outliers and can be used when distributions don't have the same total mass; entropy-regularized transport, which is robust to sampling noise and lends itself to fast computations using the Sinkhorn algorithm. This paper combines both lines of work to put robust optimal transport on solid ground. Our main contribution is a generalization of the Sinkhorn algorithm to unbalanced transport: our method alternates between the standard Sinkhorn updates and the pointwise application of a contractive function. This implies that entropic transport solvers on grid images, point clouds and sampled distributions can all be modified easily to support unbalanced transport, with a proof of linear convergence that holds in all settings. We then show how to use this method to define pseudo-distances on the full space of positive measures that satisfy key geometric axioms: (unbalanced) Sinkhorn divergences are differentiable, positive, definite, convex, statistically robust and avoid any "entropic bias" towards a shrinkage of the measures' supports.