Acknowledgement
This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Ministry of Science, ICT & Future Planning (NRF-20151009350, NRF-2016R1-A5A1008055, NRF-2016R1D1A1B03934371 and NRF-2019R1F1A1057051).
References
- C. R. Givens and R. M. Shortt, A class of wasserstein metrics for probability distributions, Mich. Math. J. 31 (1984), 231-240.
- S. Kum and S. Yun, Gradient projection methods for the n-coupling problem, J. Korean Math. Soc. 56 (2019), 1001-1016. https://doi.org/10.4134/JKMS.J180517
- M. Agueh and G. Carlier, Barycenters in the wasserstein space, SIAM J. Math. Anal. 43 (2011), 904-924. https://doi.org/10.1137/100805741
- Y.-H. Kim and B. Pass, Multi-marginal optimal transport on Riemannian manifolds, Amer. J. Math. 137 (2015), 1045-1060. https://doi.org/10.1353/ajm.2015.0024
- B. Pass, The local structure of optimal measures in the multi-marginal optimal transportation problem, Calc. Var. Partial Differential Equations 43 (2012), 529-536. https://doi.org/10.1007/s00526-011-0421-z
- G. Carlier and I. Ekeland, Matching for teams, Econ. Theory 42 (2010), 397-418. https://doi.org/10.1007/s00199-008-0415-z
- G. Carlier, A. Oberman, and E. Oudet, Numerical methods for matching for teams and wasserstein barycenters, ESAIM: M2AN 49 (2015), 1621-1642. https://doi.org/10.1051/m2an/2015033
- A. Mallasto and A. Feragen, Learning from uncertain curves: The 2-wasserstein metric for gaussian processes, In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems 30, pages 5660-5670. Curran Associates, Inc., 2017.
- J. Rabin, G. Peyre, J. Delon, and M. Bernot, Wasserstein barycenter and its application to texture mixing, In Proceedings of the Third International Conference on Scale Space and Variational Methods in Computer Vision, SSVM'11, pages 435-446, Berlin, Heidelberg, 2012.
- S. Srivastava, C. Li, and D. B. Dunson, Scalable bayes via barycenter in wasserstein space, J. Mach. Learn. Res. 19 (2018), 312-346.
- P. C. Alvarez Esteban, E. del Barrio, J. Cuesta-Albertos, and C. Matran, A fixed-point approach to barycenters in wasserstein space, J. Math. Anal. Appl. 441 (2016), 744-762. https://doi.org/10.1016/j.jmaa.2016.04.045
- R. Johnson and T. Zhang, Accelerating stochastic gradient descent using predictive variance reduction, in Adv. Neural Inf. Process. Syst. 26, NIPS'13, USA 2013.
- D. P. Bertsekas, Incremental proximal methods for large scale convex optimization, Math. Program. Ser. B 129 (2011), 163--195. https://doi.org/10.1007/s10107-011-0472-0
- L. Xiao and T. Zhang, A proximal stochastic gradient method with progressive variance reduction, SIAM J. Optim. 24 (2014), 2057--2075. https://doi.org/10.1137/140961791
- A. S. Lewis and J. Malick, Alternating projections on manifolds, Math. Oper. Res. 33 (2008), 216-234. https://doi.org/10.1287/moor.1070.0291
- R. Bhatia, T. Jain, and Y. Lim, On the bures-wasserstein distance between positive definite matrices, Expositiones Mathematicae 37 (2019), 165-191. https://doi.org/10.1016/j.exmath.2018.01.002