DOI QR코드

DOI QR Code

COMPUTATION OF DIVERGENCES AND MEDIANS IN SECOND ORDER CONES

  • Kum, Sangho (Department of Mathematics Education, College of Education Chungbuk National University)
  • 투고 : 2020.12.29
  • 심사 : 2021.07.02
  • 발행 : 2021.12.15

초록

Recently the author studied a one-parameter family of divergences and considered the related median minimization problem of finite points over these divergences in general symmetric cones. In this article, to utilize the results practically, we deal with a special symmetric cone called second order cone, which is important in optimization fields. To be more specific, concrete computations of divergences with its gradients and the unique minimizer of the median minimization problem of two points are provided skillfully.

키워드

과제정보

This work was supported by Basic Science Research Program through NRF Grant No.NRF-2017R1A2B1002008.

참고문헌

  1. S. Amari, Information geometry and its applications, Springer (Tokyo), 2016.
  2. S. Amari, Divergence function, information monotonicity and information geometry, Workshop on Information Theoretic Methods in Science and Engineering (WITMSE), 2009.
  3. S. Amari and A. Cichocki, Information geometry of divergence functions, Bull. of the Polish Acad. of Sci. Tech. Sci., 58 (2010), 183-195.
  4. R. Bhatia, T. Jain and Y. Lim, On the BuresWasserstein distance between positive definite matrices, Expo. Math., 37 (2019), 165-191. https://doi.org/10.1016/j.exmath.2018.01.002
  5. D. Dowson and B. Landau, The Frechet distance between multivariate normal distributions, J. Math. Anal., 12 (1982), 450-455.
  6. R. Frank and E. Lieb, Monotonicity of a relative R'enyi entropy, J. Math. Phys. 54 (2013), 122201. https://doi.org/10.1063/1.4838835
  7. J. Faraut and A. Koranyi, Analysis on symmetric cones, Clarendon Press, Oxford, 1994.
  8. M. Knott and C. Smith, On the optimal mapping of distributions, J. Optim. Theory Appl., 43 (1984), 39-49. https://doi.org/10.1007/BF00934745
  9. J. Kim and Y. Lim, Jordan automorphic generators of Euclidean Jordan algebras, J. Korean Math. Soc., 43 (2006), 507-528. https://doi.org/10.4134/JKMS.2006.43.3.507
  10. S. Kum, Y. Lim and S. Yun, Divergences on symmetric cones and medians, submitted for publication.
  11. Y. Lim, Geometric means on symmetric cones, Arch. der Math., 75 (2000), 39-45. https://doi.org/10.1007/s000130050471
  12. I. Olkin and F. Pukelsheim, The distance between two random vectors with given dispersion matrices, Linear Algebra Appl., 48 (1982), 257-263. https://doi.org/10.1016/0024-3795(82)90112-4
  13. S.H. Schmieta and F. Alizadeh,Extension of primal-dual interior point algorithms to symmetric cones, Math. Program. 96 (2003), 409-438. https://doi.org/10.1007/s10107-003-0380-z
  14. A. Uhlmann, Density operators as an arena for differential geometry, Rep. Math. Phys., 33 (1993), 255-263. https://doi.org/10.1016/0034-4877(93)90060-R
  15. A. Uhlmann, Transition probability (fidelity) and its relatives, Found. Phys., 41 (2011), 288-298. https://doi.org/10.1007/s10701-009-9381-y
  16. M. Wilde, A. Winter and D. Yang, Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy, Commu. Math. Phys., 331 (2014), 593-622 https://doi.org/10.1007/s00220-014-2122-x