8 Chain Rule Optimal Transport
217
52. Pitrik, J., Virosztek, D.: On the joint convexity of the Bregman divergence of matrices. Lett.
Math. Phys. 105(5), 675–692 (2015)
53. Reynolds, D.A., Quatieri, T.F., Dunn, R.B.: Speaker verification using adapted Gaussian mixture models. Digital Signal Process. 10(1–3), 19–41 (2000)
54. Rubner, Y., Tomasi, C., Guibas, L.J.: The earth mover’s distance as a metric for image retrieval.
Int. J. Comput. Vis. 40(2), 99–121 (2000)
55. Rüschendorf, L.: The Wasserstein distance and approximation theorems. Probab. Theory Relat.
Fields 70, 117–129 (1985)
56. Santambrogio, F.: Optimal Transport for Applied Mathematicians, pp. 99–102. Birkäuser, NY
(2015)
57. Schwander, O., Nielsen, F.: Learning mixtures by simplifying kernel density estimators. In:
Matrix Information Geometry, pp. 403–426. Springer (2013)
58. Silva, J., Narayanan, S.: Upper bound Kullback-Leibler divergence for hidden Markov models
with application as discrimination measure for speech recognition. In: IEEE International
Symposium on Information Theory (ISIT), pp. 2299–2303. IEEE (2006)
59. Singer, Y., Warmuth, M.K.: Batch and on-line parameter estimation of Gaussian mixtures based
on the joint entropy. In: NIPS 578–584 (1999)
60. Takatsu, A., et al.: Wasserstein geometry of Gaussian measures. Osaka J. Math. 48(4), 1005–
1026 (2011)
61. Van Erven, T., Harremos, P.: Rényi divergence and Kullback-Leibler divergence. IEEE Trans.
Inf. Theory 60(7), 3797–3820 (2014)
62. Vaserstein, L.N.: Markov processes over denumerable products of spaces, describing large
systems of automata. Probl. Peredachi Informatsii 5(3), 64–72 (1969)
63. Vigelis, R.F., De Andrade, L.H., Cavalcante, C.C.: Properties of a generalized divergence
related to Tsallis generalized divergence. IEEE Trans. Inf. Theory 66(5), 2891–2897 (2019)
64. Xiao, H., Rasul, K., Vollgraf, R.: Fashion-MNIST: a novel image dataset for benchmarking
machine learning algorithms. Technical report, Zalando Research, Berlin, Germany (2017).
arXiv:cs.LG/1708.07747
65. Xie, L., Ugrinovskii, V.A., Petersen, I.R.: Probabilistic distances between finite-state finitealphabet hidden Markov models. IEEE Trans. Autom. Control. 50(4), 505–511 (2005)
217
52. Pitrik, J., Virosztek, D.: On the joint convexity of the Bregman divergence of matrices. Lett.
Math. Phys. 105(5), 675–692 (2015)
53. Reynolds, D.A., Quatieri, T.F., Dunn, R.B.: Speaker verification using adapted Gaussian mixture models. Digital Signal Process. 10(1–3), 19–41 (2000)
54. Rubner, Y., Tomasi, C., Guibas, L.J.: The earth mover’s distance as a metric for image retrieval.
Int. J. Comput. Vis. 40(2), 99–121 (2000)
55. Rüschendorf, L.: The Wasserstein distance and approximation theorems. Probab. Theory Relat.
Fields 70, 117–129 (1985)
56. Santambrogio, F.: Optimal Transport for Applied Mathematicians, pp. 99–102. Birkäuser, NY
(2015)
57. Schwander, O., Nielsen, F.: Learning mixtures by simplifying kernel density estimators. In:
Matrix Information Geometry, pp. 403–426. Springer (2013)
58. Silva, J., Narayanan, S.: Upper bound Kullback-Leibler divergence for hidden Markov models
with application as discrimination measure for speech recognition. In: IEEE International
Symposium on Information Theory (ISIT), pp. 2299–2303. IEEE (2006)
59. Singer, Y., Warmuth, M.K.: Batch and on-line parameter estimation of Gaussian mixtures based
on the joint entropy. In: NIPS 578–584 (1999)
60. Takatsu, A., et al.: Wasserstein geometry of Gaussian measures. Osaka J. Math. 48(4), 1005–
1026 (2011)
61. Van Erven, T., Harremos, P.: Rényi divergence and Kullback-Leibler divergence. IEEE Trans.
Inf. Theory 60(7), 3797–3820 (2014)
62. Vaserstein, L.N.: Markov processes over denumerable products of spaces, describing large
systems of automata. Probl. Peredachi Informatsii 5(3), 64–72 (1969)
63. Vigelis, R.F., De Andrade, L.H., Cavalcante, C.C.: Properties of a generalized divergence
related to Tsallis generalized divergence. IEEE Trans. Inf. Theory 66(5), 2891–2897 (2019)
64. Xiao, H., Rasul, K., Vollgraf, R.: Fashion-MNIST: a novel image dataset for benchmarking
machine learning algorithms. Technical report, Zalando Research, Berlin, Germany (2017).
arXiv:cs.LG/1708.07747
65. Xie, L., Ugrinovskii, V.A., Petersen, I.R.: Probabilistic distances between finite-state finitealphabet hidden Markov models. IEEE Trans. Autom. Control. 50(4), 505–511 (2005)