31
Views
0
CrossRef citations to date
0
Altmetric
Research Article

On the Wasserstein Median of Probability Measures

, &
Received 17 Mar 2023, Accepted 13 Jun 2024, Accepted author version posted online: 01 Jul 2024
Accepted author version

References

  • Afsari, B. (2011). Riemannian Lp center of mass: Existence, uniqueness, and convexity, Proceedings of the American Mathematical Society 139(02): 655–655.
  • Agueh, M. and Carlier, G. (2011). Barycenters in the Wasserstein Space, SIAM Journal on Mathematical Analysis 43(2): 904–924.
  • Altschuler, J., Chewi, S., Gerber, P. R. and Stromme, A. (2021). Averaging on the Bures-Wasserstein manifold: Dimension-free convergence of gradient descent, in M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang and J. W. Vaughan (eds), Advances in Neural Information Processing Systems, Vol. 34, Curran Associates, Inc., pp. 22132–22145.
  • Álvarez-Esteban, P. C., del Barrio, E., Cuesta-Albertos, J. and Matrán, C. (2016). A fixed-point approach to barycenters in Wasserstein space, Journal of Mathematical Analysis and Applications 441(2): 744–762.
  • Ambrosio, L., Brué, E. and Semola, D. (2021). Lectures on Optimal Transport, number volume 130 in Unitext - La Matematica per Il 3 + 2, Springer, Cham.
  • Ambrosio, L., Caffarelli, L. A. and Salsa, S. (eds) (2003). Optimal Transportation and Applications: Lectures given at the C.I.M.E. Summer School Held in Martina Franca, Italy, September 2-8, 2001, number 1813 in Lecture Notes in Mathematics, Springer, Berlin; New York.
  • Ambrosio, L., Gigli, N. and Savaré, G. (2005). Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser, Boston.
  • Arjovsky, M., Chintala, S. and Bottou, L. (2017). Wasserstein generative adversarial networks, in D. Precup and Y. W. Teh (eds), Proceedings of the 34th International Conference on Machine Learning, Vol. 70 of Proceedings of Machine Learning Research, PMLR, pp. 214–223.
  • Bernton, E., Jacob, P. E., Gerber, M. and Robert, C. P. (2019a). Approximate Bayesian computation with the Wasserstein distance, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 81(2): 235–269.
  • Bernton, E., Jacob, P. E., Gerber, M. and Robert, C. P. (2019b). On parameter estimation with the Wasserstein distance, Information and Inference: A Journal of the IMA 8(4): 657–676.
  • Bhattacharya, A. and Bhattacharya, R. (2012). Nonparametric Inference on Manifolds: With Applications to Shape Spaces, Cambridge University Press, Cambridge.
  • Bigot, J. and Klein, T. (2018). Characterization of barycenters in the Wasserstein space by averaging optimal transport maps, ESAIM: Probability and Statistics 22: 35–57.
  • Chartrand, R. and Yin, W. (2008). Iteratively reweighted algorithms for compressive sensing, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, IEEE, Las Vegas, NV, USA, pp. 3869–3872.
  • Claici, S., Chien, E. and Solomon, J. (2018). Stochastic Wasserstein barycenters, in J. Dy and A. Krause (eds), Proceedings of the 35th International Conference on Machine Learning, Vol. 80 of Proceedings of Machine Learning Research, PMLR, pp. 999–1008.
  • Courty, N., Flamary, R., Habrard, A. and Rakotomamonjy, A. (2017). Joint distribution optimal transportation for domain adaptation, in I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan and R. Garnett (eds), Advances in Neural Information Processing Systems, Vol. 30, Curran Associates, Inc.
  • Courty, N., Flamary, R., Tuia, D. and Rakotomamonjy, A. (2017). Optimal Transport for Domain Adaptation, IEEE Transactions on Pattern Analysis and Machine Intelligence 39(9): 1853–1865.
  • Cuturi, M. (2013). Sinkhorn distances: Lightspeed computation of optimal transport, in C. Burges, L. Bottou, M. Welling, Z. Ghahramani and K. Weinberger (eds), Advances in Neural Information Processing Systems, Vol. 26, Curran Associates, Inc.
  • Cuturi, M. and Doucet, A. (2014). Fast computation of wasserstein barycenters, in E. P. Xing and T. Jebara (eds), Proceedings of the 31st International Conference on Machine Learning, Vol. 32 of Proceedings of Machine Learning Research, PMLR, Bejing, China, pp. 685–693.
  • Daubechies, I., DeVore, R., Fornasier, M. and Güntürk, C. S. (2010). Iteratively reweighted least squares minimization for sparse recovery, Communications on Pure and Applied Mathematics 63(1): 1–38.
  • del Barrio, E., Giné, E. and Matrán, C. (1999). Central Limit Theorems for the Wasserstein Distance Between the Empirical and the True Distributions, The Annals of Probability 27(2).
  • Dowson, D. and Landau, B. (1982). The Fréchet distance between multivariate normal distributions, Journal of Multivariate Analysis 12(3): 450–455.
  • Dvurechenskii, P., Dvinskikh, D., Gasnikov, A., Uribe, C. and Nedich, A. (2018). Decentralize and randomize: Faster algorithm for wasserstein barycenters, in S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi and R. Garnett (eds), Advances in Neural Information Processing Systems, Vol. 31, Curran Associates, Inc.
  • El Moselhy, T. A. and Marzouk, Y. M. (2012). Bayesian inference with optimal maps, Journal of Computational Physics 231(23): 7815–7850.
  • Fletcher, P. T., Venkatasubramanian, S. and Joshi, S. (2009). The geometric median on Riemannian manifolds with application to robust atlas estimation, NeuroImage 45(1): S143–S152.
  • Fournier, N. and Guillin, A. (2015). On the rate of convergence in Wasserstein distance of the empirical measure, Probability Theory and Related Fields 162(3-4): 707–738.
  • Givens, C. R. and Shortt, R. M. (1984). A class of Wasserstein metrics for probability distributions., Michigan Mathematical Journal 31(2).
  • Gorodnitsky, I. and Rao, B. (1997). Sparse signal reconstruction from limited data using FOCUSS: A re-weighted minimum norm algorithm, IEEE Transactions on Signal Processing 45(3): 600–616.
  • Hampel, F. R. (1971). A General Qualitative Definition of Robustness, The Annals of Mathematical Statistics 42(6): 1887–1896.
  • Huber, P. J. (1981). Robust Statistics, Wiley Series in Probability and Mathematical Statistics, Wiley, New York.
  • Hütter, J.-C. and Rigollet, P. (2021). Minimax estimation of smooth optimal transport maps, The Annals of Statistics 49(2).
  • Kantorovitch, L. (1958). On the Translocation of Masses, Management Science 5(1): 1–4.
  • Kaufman, L. and Rousseeuw, P. J. (1990). Partitioning Around Medoids (Program PAM), Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., Hoboken, NJ, USA, pp. 68–125.
  • Kendall, W. S. (1990). Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence, Proceedings of the London Mathematical Society s3-61(2): 371–406.
  • Knott, M. and Smith, C. S. (1984). On the optimal mapping of distributions, Journal of Optimization Theory and Applications 43(1): 39–49.
  • Kolouri, S., Zou, Y. and Rohde, G. K. (2016). Sliced wasserstein kernels for probability distributions, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR).
  • Korotin, A., Li, L., Solomon, J. and Burnaev, E. (2021). Continuous wasserstein-2 barycenter estimation without minimax optimization, International Conference on Learning Representations.
  • Lawson, C. L. (1961). Contributions to the Theory of Linear Least Maximum Approximation, PhD thesis, University of California Los Angeles.
  • Le Gouic, T. and Loubes, J.-M. (2017). Existence and consistency of Wasserstein barycenters, Probability Theory and Related Fields 168(3-4): 901–917.
  • LeCun, Y., Cortes, C. and Burges, C. (1998). The MNIST Database of Handwritten Digits, http://yann.lecun.com/exdb/mnist/.
  • Li, L., Genevay, A., Yurochkin, M. and Solomon, J. M. (2020). Continuous regularized wasserstein barycenters, in H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan and H. Lin (eds), Advances in Neural Information Processing Systems, Vol. 33, Curran Associates, Inc., pp. 17755–17765.
  • MacQueen, J. B. (1967). Some Methods for Classification and Analysis of Multivariate Observations, in L. M. L. Cam and J. Neyman (eds), Proc. of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, pp. 281–297.
  • Malagò, L., Montrucchio, L. and Pistone, G. (2018). Wasserstein Riemannian geometry of Gaussian densities, Information Geometry 1(2): 137–179.
  • Manning, C. D., Raghavan, P. and Schütze, H. (2008). Introduction to Information Retrieval, Cambridge University Press, New York.
  • Manole, T., Balakrishnan, S., Niles-Weed, J. and Wasserman, L. (2021). Plugin Estimation of Smooth Optimal Transport Maps.
  • McCann, R. J. (1997). A Convexity Principle for Interacting Gases, Advances in Mathematics 128(1): 153–179.
  • McCullagh, P. and Nelder, J. A. (1998). Generalized Linear Models, number 37 in Monographs on Statistics and Applied Probability, 2nd ed edn, Chapman & Hall/CRC, Boca Raton.
  • Minsker, S. (2015). Geometric median and robust estimation in Banach spaces, Bernoulli 21(4).
  • Monge, G. (1781). Mémoire Sur La Théorie Des Déblais et Des Remblais, De l’Imprimerie Royale.
  • Nelder, J. A. and Wedderburn, R. W. M. (1972). Generalized Linear Models, Journal of the Royal Statistical Society. Series A (General) 135(3): 370.
  • Newcomb, S. (1891). Measures of the Velocity of Light Made Under the Direction of the Secretary of the Navy During the Years 1880-1882, Vol. 2 of Astronomical Papers of the United States Naval Observatory 1882-1986, Bureau of Navigation, Navy Department, Washington, D.C.
  • Olkin, I. and Pukelsheim, F. (1982). The distance between two random vectors with given dispersion matrices, Linear Algebra and its Applications 48: 257–263.
  • Osborne, M. R. (1985). Finite Algorithms in Optimization and Data Analysis, Wiley Series in Probability and Mathematical Statistics, Wiley, Chichester; New York.
  • Otto, F. (2001). The geometry of dissipative evolution equations: the porous medium equation, Communications in Partial Differential Equations 26(1-2): 101–174.
  • Panaretos, V. M. and Zemel, Y. (2020). An Invitation to Statistics in Wasserstein Space, SpringerBriefs in Probability and Mathematical Statistics, Springer International Publishing, Cham.
  • Pennec, X. (2006). Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements, Journal of Mathematical Imaging and Vision 25(1): 127–154.
  • Peyré, G. and Cuturi, M. (2019). Computational Optimal Transport: With Applications to Data Science, Foundations and Trends[textregistered] in Machine Learning 11(5-6): 355–607.
  • Preston, S. H., Heuveline, P. and Guillot, M. (2009). Demography: Measuring and Modeling Population Processes, nachdr. edn, Blackwell, Oxford.
  • Ramdas, A., Trillos, N. and Cuturi, M. (2017). On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests, Entropy 19(2): 47.
  • Rice, J. R. (1964). The Approximations of Functions, Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Pub. Co, Reading, Mass.
  • Rousseeuw, P. J. (1987). Silhouettes: A graphical aid to the interpretation and validation of cluster analysis, Journal of Computational and Applied Mathematics 20: 53–65.
  • Rousseeuw, P. J. and Leroy, A. M. (1987). Robust Regression and Outlier Detection, Wiley, New York.
  • Rüschendorf, L. and Uckelmann, L. (2002). On the n-Coupling Problem, Journal of Multivariate Analysis 81(2): 242–258.
  • Sammut, C. and Webb, G. I. (2011). Encyclopedia of Machine Learning, Springer Reference, Springer, New York.
  • Srivastava, S., Li, C. and Dunson, D. B. (2018). Scalable bayes via barycenter in wasserstein space, Journal of Machine Learning Research 19(1): 312–346.
  • Takatsu, A. (2011). Wasserstein geometry of Gaussian measures, Osaka Journal of Mathematics 48(4): 1005–1026.
  • Vardi, Y. and Zhang, C.-H. (2000). The multivariate L1-median and associated data depth, Proceedings of the National Academy of Sciences 97(4): 1423–1426.
  • Villani, C. (2003). Topics in Optimal Transportation, Vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, S.l.
  • Villani, C. (2009). Optimal Transport: Old and New, number 338 in Grundlehren Der Mathematischen Wissenschaften, Springer, Berlin.
  • Weiszfeld, E. (1937). Sur le point pour lequel la Somme des distances de n points donnes est minimum, Tohoku Mathematical Journal, First Series 43: 355–386.
  • Xie, Y., Wang, X., Wang, R. and Zha, H. (2020). A fast proximal point method for computing exact wasserstein distance, in R. P. Adams and V. Gogate (eds), Proceedings of the 35th Uncertainty in Artificial Intelligence Conference, Vol. 115 of Proceedings of Machine Learning Research, PMLR, pp. 433–453.
  • You, K. and Suh, C. (2022). Parameter estimation and model-based clustering with spherical normal distribution on the unit hypersphere, Computational Statistics & Data Analysis 171: 107457.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.