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Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures

arXiv 1508.07941 ↗First posted 31 Aug 2015math.OCcited by 138

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Abstract (arXiv)

We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, that quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.

Key facts. Cited by 138 papers in the map (data through 7 Oct 2026). First posted on arXiv 31 Aug 2015.

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Citing papers per year: 2021: 14; 2022: 22; 2023: 25; 2024: 21; 2025: 28; 2026: 28.

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  1. 2021Unbalanced minibatch Optimal Transport; applications to Domain Adaptationcited by 123
  2. 2022Action Matching: Learning Stochastic Dynamics from Samplescited by 94
  3. 2025Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Expertscited by 85
  4. 2022Variational inference via Wasserstein gradient flowscited by 84
  5. 2023Recent Advances in Optimal Transport for Machine Learningcited by 66
  6. 2022Unbalanced Optimal Transport, from Theory to Numericscited by 45
  7. 2023A Computational Framework for Solving Wasserstein Lagrangian Flowscited by 43
  8. 2021Unbalanced Feature Transport for Exemplar-based Image Translationcited by 41
  9. 2023Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transportcited by 39
  10. 2021Accurate Point Cloud Registration with Robust Optimal Transportcited by 38
  11. 2023Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translationcited by 32
  12. 2021Towards a mathematical theory of trajectory inferencecited by 31
38 more
  1. 2021Unbalanced Optimal Transport through Non-negative Penalized Linear Regressioncited by 31
  2. 2022Birth-death dynamics for sampling: Global convergence, approximations and their asymptoticscited by 30
  3. 2021Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysiscited by 27
  4. 2023Scalable Optimal Transport Methods in Machine Learning: A Contemporary Surveycited by 27
  5. 2023GENOT: Entropic (Gromov) Wasserstein Flow Matching with Applications to Single-Cell Genomicscited by 27
  6. 2023Learning Gaussian Mixtures Using the Wasserstein-Fisher-Rao Gradient Flowcited by 25
  7. 2022Aligning individual brains with Fused Unbalanced Gromov-Wassersteincited by 23
  8. 2022Faster Unbalanced Optimal Transport: Translation invariant Sinkhorn and 1-D Frank-Wolfecited by 22
  9. 2024Scalable Wasserstein Gradient Flow for Generative Modeling through Unbalanced Optimal Transportcited by 22
  10. 2023Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein spacecited by 20
  11. 2025Joint Velocity-Growth Flow Matching for Single-Cell Dynamics Modelingcited by 20
  12. 2022Sliced Optimal Partial Transportcited by 19
  13. 2021Sparse Needlets for Lighting Estimation with Spherical Transport Losscited by 18
  14. 2025Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approachcited by 18
  15. 2022VMRF: View Matching Neural Radiance Fieldscited by 17
  16. 2022On Unbalanced Optimal Transport: Gradient Methods, Sparsity and Approximation Errorcited by 16
  17. 2022Neural Unbalanced Optimal Transport via Cycle-Consistent Semi-Couplingscited by 16
  18. 2023An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flowcited by 16
  19. 2024Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spacescited by 15
  20. 2022Weisfeiler-Lehman meets Gromov-Wassersteincited by 14
  21. 2022An Exponentially Converging Particle Method for the Mixed Nash Equilibrium of Continuous Gamescited by 14
  22. 2021Entropy Partial Transport with Tree Metrics: Theory and Practicecited by 13
  23. 2024P2OT: Progressive Partial Optimal Transport for Deep Imbalanced Clusteringcited by 13
  24. 2024Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalitiescited by 13
  25. 2024Stein transport for Bayesian inferencecited by 11
  26. 2022Chained Generalisation Boundscited by 10
  27. 2023Scalable Unbalanced Sobolev Transport for Measures on a Graphcited by 10
  28. 2025Weighted quantization using MMD: From mean field to mean shift via gradient flowscited by 10
  29. 2023Unbalanced Diffusion Schr"odinger Bridgecited by 9
  30. 2024Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transportcited by 9
  31. 2021The Gene Mover's Distance: Single-cell similarity via Optimal Transportcited by 8
  32. 2023Robust Estimation under the Wasserstein Distancecited by 8
  33. 2023Linear Optimal Partial Transport Embeddingcited by 8
  34. 2023Light Unbalanced Optimal Transportcited by 8
  35. 2023Wasserstein Gradient Flows for Optimizing Gaussian Mixture Policiescited by 8
  36. 2023Analyzing and Improving Optimal-Transport-based Adversarial Networkscited by 8
  37. 2025OTSurv: A Novel Multiple Instance Learning Framework for Survival Prediction with Heterogeneity-aware Optimal Transportcited by 8
  38. 2022Riemannian block SPD coupling manifold and its application to optimal transportcited by 7

50 most cited of 138 All 138 (JSON)