216
F. Nielsen and K. Sun
25. Hershey, J.R., Olsen, P.A.: Approximating the Kullback-Leibler divergence between Gaussian
mixture models. In: ICASSP, vol. 4, pp. IV–317. IEEE (2007)
26. Kantorovich, L.: On the transfer of masses. Doklady Akademii Nauk 37(2), 227–229 (1942).
(in Russian)
27. Kantorovitch, L.: On the translocation of masses. Manag. Sci. 5(1), 1–4 (1958)
28. Khosravifard, M., Fooladivanda, D., Gulliver, T.A.: Confliction of the convexity and metric
properties in f -divergences. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 90(9),
1848–1853 (2007)
29. Kingma, D.P., Welling, M.: Auto-encoding variational Bayes. In: ICLR (2014)
30. Komaki, F.: Bayesian prediction based on a class of shrinkage priors for location-scale models.
Ann. Inst. Stat. Math. 59(1), 135–146 (2007)
31. Korte, B., Vygen, J.: Linear programming algorithms. In: Combinatorial Optimization, pp.
75–102. Springer (2018)
32. LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document
recognition. Proc. IEEE 86(11), 2278–2324 (1998)
33. Liu, Z., Huang, Q.: A new distance measure for probability distribution function of mixture
type. In: ICASSP, vol. 1, pp. 616–619. IEEE (2000)
34. Monge, G.: Mémoire sur la théorie des déblais et des remblais. Imprimerie Royale (1781)
35. Nielsen, F.: A family of statistical symmetric divergences based on Jensen’s inequality (2010).
arXiv:1009.4004
36. Nielsen, F.: Closed-form information-theoretic divergences for statistical mixtures. In: 2012
21st International Conference on Pattern Recognition (ICPR), pp. 1723–1726. IEEE (2012)
37. Nielsen, F.: Generalized Bhattacharyya and Chernoff upper bounds on bayes error using quasiarithmetic means. Pattern Recognit. Lett. 42, 25–34 (2014)
38. Nielsen, F.: The statistical Minkowski distances: closed-form formula for Gaussian mixture
models (2019). arXiv:1901.03732
39. Nielsen, F., Garcia, V.: Statistical exponential families: a digest with flash cards (2009).
arXiv:0911.4863
40. Nielsen, F., Nock, R.: A closed-form expression for the Sharma-Mittal entropy of exponential
families. J. Phys. A: Math. Theor. 45(3), 032003 (2011)
41. Nielsen, F., Nock, R.: On Rényi and Tsallis entropies and divergences for exponential families
(2011). arXiv:1105.3259
42. Nielsen, F., Nock, R.: On the chi square and higher-order chi distances for approximating
f -divergences. IEEE Signal Process. Lett. 21(1), 10–13 (2014)
43. Nielsen, F., Nock, R.: On w-mixtures: finite convex combinations of prescribed component
distributions (2017). CoRR arXiv:abs/1708.00568
44. Nielsen, F., Sun, K.: Guaranteed bounds on information-theoretic measures of univariate mixtures using piecewise log-sum-exp inequalities. Entropy 18(12), 442 (2016)
45. Nielsen, F., Sun, K.: Guaranteed deterministic bounds on the total variation distance between
univariate mixtures. IEEE Mach. Learn. Signal Process. (MLSP) 1–6 (2018)
46. Nielsen, M.A., Chuang, I.: Quantum computation and quantum information (2002)
47. Österreicher, F., Vajda, I.: A new class of metric divergences on probability spaces and its
applicability in statistics. Ann. Inst. Stat. Math. 55(3), 639–653 (2003)
48. Ozawa, R., Yokota, T.: Stability of RCD condition under concentration topology. J. Phys. A:
Math. Theor. 45(3), 032003 (2011)
49. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M.,
Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. J.
Mach. Learn. Res. 12, 2825–2830 (2011)
50. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M.,
Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: machine learning in Python. J.
Mach. Learn. Res. 12, 2825–2830 (2011)
51. Peyré, G., Cuturi, M., et al.: Computational optimal transport. Found. Trends® in Mach. Learn.
11(5–6), 355–607 (2019)
F. Nielsen and K. Sun
25. Hershey, J.R., Olsen, P.A.: Approximating the Kullback-Leibler divergence between Gaussian
mixture models. In: ICASSP, vol. 4, pp. IV–317. IEEE (2007)
26. Kantorovich, L.: On the transfer of masses. Doklady Akademii Nauk 37(2), 227–229 (1942).
(in Russian)
27. Kantorovitch, L.: On the translocation of masses. Manag. Sci. 5(1), 1–4 (1958)
28. Khosravifard, M., Fooladivanda, D., Gulliver, T.A.: Confliction of the convexity and metric
properties in f -divergences. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 90(9),
1848–1853 (2007)
29. Kingma, D.P., Welling, M.: Auto-encoding variational Bayes. In: ICLR (2014)
30. Komaki, F.: Bayesian prediction based on a class of shrinkage priors for location-scale models.
Ann. Inst. Stat. Math. 59(1), 135–146 (2007)
31. Korte, B., Vygen, J.: Linear programming algorithms. In: Combinatorial Optimization, pp.
75–102. Springer (2018)
32. LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document
recognition. Proc. IEEE 86(11), 2278–2324 (1998)
33. Liu, Z., Huang, Q.: A new distance measure for probability distribution function of mixture
type. In: ICASSP, vol. 1, pp. 616–619. IEEE (2000)
34. Monge, G.: Mémoire sur la théorie des déblais et des remblais. Imprimerie Royale (1781)
35. Nielsen, F.: A family of statistical symmetric divergences based on Jensen’s inequality (2010).
arXiv:1009.4004
36. Nielsen, F.: Closed-form information-theoretic divergences for statistical mixtures. In: 2012
21st International Conference on Pattern Recognition (ICPR), pp. 1723–1726. IEEE (2012)
37. Nielsen, F.: Generalized Bhattacharyya and Chernoff upper bounds on bayes error using quasiarithmetic means. Pattern Recognit. Lett. 42, 25–34 (2014)
38. Nielsen, F.: The statistical Minkowski distances: closed-form formula for Gaussian mixture
models (2019). arXiv:1901.03732
39. Nielsen, F., Garcia, V.: Statistical exponential families: a digest with flash cards (2009).
arXiv:0911.4863
40. Nielsen, F., Nock, R.: A closed-form expression for the Sharma-Mittal entropy of exponential
families. J. Phys. A: Math. Theor. 45(3), 032003 (2011)
41. Nielsen, F., Nock, R.: On Rényi and Tsallis entropies and divergences for exponential families
(2011). arXiv:1105.3259
42. Nielsen, F., Nock, R.: On the chi square and higher-order chi distances for approximating
f -divergences. IEEE Signal Process. Lett. 21(1), 10–13 (2014)
43. Nielsen, F., Nock, R.: On w-mixtures: finite convex combinations of prescribed component
distributions (2017). CoRR arXiv:abs/1708.00568
44. Nielsen, F., Sun, K.: Guaranteed bounds on information-theoretic measures of univariate mixtures using piecewise log-sum-exp inequalities. Entropy 18(12), 442 (2016)
45. Nielsen, F., Sun, K.: Guaranteed deterministic bounds on the total variation distance between
univariate mixtures. IEEE Mach. Learn. Signal Process. (MLSP) 1–6 (2018)
46. Nielsen, M.A., Chuang, I.: Quantum computation and quantum information (2002)
47. Österreicher, F., Vajda, I.: A new class of metric divergences on probability spaces and its
applicability in statistics. Ann. Inst. Stat. Math. 55(3), 639–653 (2003)
48. Ozawa, R., Yokota, T.: Stability of RCD condition under concentration topology. J. Phys. A:
Math. Theor. 45(3), 032003 (2011)
49. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M.,
Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: Machine learning in python. J.
Mach. Learn. Res. 12, 2825–2830 (2011)
50. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M.,
Prettenhofer, P., Weiss, R., Dubourg, V., et al.: Scikit-learn: machine learning in Python. J.
Mach. Learn. Res. 12, 2825–2830 (2011)
51. Peyré, G., Cuturi, M., et al.: Computational optimal transport. Found. Trends® in Mach. Learn.
11(5–6), 355–607 (2019)