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References

The literature behind each part of the package. The first section is the work it builds on.

This package

  • D. Gentile, J. M. Murphy. Static and dynamic approaches to computing barycenters of probability measures on graphs. arXiv:2603.26940, 2026. https://arxiv.org/abs/2603.26940
  • D. Gentile, J. M. Murphy. Regularized 2-Wasserstein barycenters on discrete metric spaces. 2025 International Conference on Sampling Theory and Applications (SampTA), IEEE, 2025, pp. 1–5.

The discrete transport metric

  • J. Maas. Gradient flows of the entropy for finite Markov chains. Journal of Functional Analysis 261 (8), 2011, 2250–2292.
  • S.-N. Chow, W. Huang, Y. Li, H. Zhou. Fokker–Planck equations for a free energy functional or Markov process on a graph. Archive for Rational Mechanics and Analysis 203 (3), 2012, 969–1008.
  • A. Mielke. Geodesic convexity of the relative entropy in reversible Markov chains. Calculus of Variations and Partial Differential Equations 48, 2013, 1–31.
  • M. Erbar, J. Maas. Ricci curvature of finite Markov chains via convexity of the entropy. Archive for Rational Mechanics and Analysis 206 (3), 2012, 997–1038.
  • M. Erbar, J. Maas, M. Wirth. On the geometry of geodesics in discrete optimal transport. Calculus of Variations and Partial Differential Equations 58 (1), 2019, 19.
  • N. Gigli, J. Maas. Gromov–Hausdorff convergence of discrete transportation metrics. SIAM Journal on Mathematical Analysis 45 (2), 2013, 879–899.
  • P. Gladbach, E. Kopfer, J. Maas. Scaling limits of discrete optimal transport. SIAM Journal on Mathematical Analysis 52 (3), 2020, 2759–2802.
  • J. Solomon, R. Rustamov, L. Guibas, A. Butscher. Continuous-flow graph transportation distances. arXiv:1603.06927, 2016.

Computing geodesics

  • J.-D. Benamou, Y. Brenier. A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numerische Mathematik 84 (3), 2000, 375–393.
  • M. Erbar, M. Rumpf, B. Schmitzer, S. Simon. Computation of optimal transport on discrete metric measure spaces. Numerische Mathematik 144 (1), 2020, 157–200.
  • P. Goulart, Y. Chen. Clarabel: An interior-point solver for conic programs with quadratic objectives. arXiv:2405.12762, 2024.
  • S. Diamond, S. Boyd. CVXPY: A Python-embedded modeling language for convex optimization. Journal of Machine Learning Research 17 (83), 2016, 1–5.
  • A. Paszke et al. PyTorch: An imperative style, high-performance deep learning library. Advances in Neural Information Processing Systems 32, 2019.

Barycenters and barycentric coordinates

  • M. Agueh, G. Carlier. Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis 43 (2), 2011, 904–924.
  • Y.-H. Kim, B. Pass. Wasserstein barycenters over Riemannian manifolds. Advances in Mathematics 307, 2017, 640–683.
  • P. C. Álvarez-Esteban, E. del Barrio, J. A. Cuesta-Albertos, C. Matrán. A fixed-point approach to barycenters in Wasserstein space. Journal of Mathematical Analysis and Applications 441 (2), 2016, 744–762.
  • J. M. Altschuler, E. Boix-Adsera. Wasserstein barycenters are NP-hard to compute. SIAM Journal on Mathematics of Data Science 4 (1), 2022, 179–203.
  • M. Werenski, R. Jiang, A. Tasissa, S. Aeron, J. M. Murphy. Measure estimation in the barycentric coding model. Proceedings of the 39th International Conference on Machine Learning, PMLR 162, 2022, pp. 23781–23803.
  • M. Werenski, B. Mallery, S. Aeron, J. M. Murphy. Linearized Wasserstein barycenters: synthesis, analysis, representational capacity, and applications. Proceedings of the 28th International Conference on Artificial Intelligence and Statistics, PMLR 258, 2025, pp. 4555–4563.
  • B. Mallery, J. M. Murphy, S. Aeron. Synthesis and analysis of data as probability measures with entropy-regularized optimal transport. Proceedings of the 28th International Conference on Artificial Intelligence and Statistics, PMLR 258, 2025, pp. 2584–2592.

Entropic transport

  • R. Sinkhorn, P. Knopp. Concerning nonnegative matrices and doubly stochastic matrices. Pacific Journal of Mathematics 21 (2), 1967, 343–348.
  • M. Cuturi. Sinkhorn distances: lightspeed computation of optimal transport. Advances in Neural Information Processing Systems 26, 2013.
  • J.-D. Benamou, G. Carlier, M. Cuturi, L. Nenna, G. Peyré. Iterative Bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing 37 (2), 2015, A1111–A1138.
  • N. Bonneel, G. Peyré, M. Cuturi. Wasserstein barycentric coordinates: histogram regression using optimal transport. ACM Transactions on Graphics 35 (4), 2016, Article 71.
  • H. Janati, M. Cuturi, A. Gramfort. Debiased Sinkhorn barycenters. Proceedings of the 37th International Conference on Machine Learning, PMLR 119, 2020, pp. 4692–4701.
  • G. Peyré, M. Cuturi. Computational optimal transport. Foundations and Trends in Machine Learning 11 (5–6), 2019, 355–607.

Background

  • R. Jordan, D. Kinderlehrer, F. Otto. The variational formulation of the Fokker–Planck equation. SIAM Journal on Mathematical Analysis 29 (1), 1998, 1–17.
  • L. Ambrosio, N. Gigli, G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser, 2005.
  • C. Villani. Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften 338, Springer, 2009.
  • F. Santambrogio. Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and Their Applications 87, Birkhäuser, 2015.