Information geometry and its applications: Convex function and dually flat manifold

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 5416, Issue , 2009, Pages 75-102

  Amari, Shun Ichi ^a  

^a RIKEN BRAIN SCIENCE INSTITUTE   (Japan)

Author keywords

Convex function; Dual a new connections; dually at manifold; generalized pythagorean theorem; Information geometry; Legendre transformation; Riemannian geometry

Indexed keywords

CONVEX FUNCTION; DUAL A NEW CONNECTIONS; DUALLY AT MANIFOLD; GENERALIZED PYTHAGOREAN THEOREM; INFORMATION GEOMETRY; LEGENDRE TRANSFORMATION; RIEMANNIAN GEOMETRY;

COMPUTATIONAL GEOMETRY;

PROBABILITY DISTRIBUTIONS;

EID: 67649967097     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-642-00826-9_4     Document Type: Conference Paper

References (28)

1. Amari, S.: Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics, vol. 28. Springer, Heidelberg (1985)
2. Amari, S.: Differential geometry of a parametric family of invertible linear systems-Riemannian metric, dual affine connections and divergence. Mathematical Systems Theory 20, 53-82 (1987)
3. Amari, S.: Information geometry of the EM and em algorithms for neural networks. Neural Networks 8-9, 1379-1408 (1995) (Pubitemid 26072898)
4. Amari, S.: Natural gradient works e.ciently in learning. Neural Computation 10, 251-276 (1998) (Pubitemid 128463152)
5. Amari, S.: Information geometry on hierarchy of probability distributions. IEEE Transactions on Information Theory 47, 1701-1711 (2001)
6. Amari, S., Kawanabe, M.: Information geometry of estimating functions in semi parametric statistical models. Bernoulli. 3(1), 29-54 (1997)
7. Amari, S., Kurata, K., Nagaoka, H.: Information geometry of Boltzmann machines. IEEE Transactions on Neural Networks 3, 260-271 (1992)
8. Amari, S., Nagaoka, H.: Methods of Information Geometry. Translations of Mathematical Monographs, vol. 191. AMS & Oxford University Press (2000)
9. Amari, S., Park, H., Ozeki, T.: Singularities affect dynamics of learning in neuromanifolds. Neural Computation 18, 1007-1065 (2006)
10. Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Physics 7, 200-217 (1967)
11. Byrne, W.: Alternating minimization and Boltzmann machine learning. IEEE Transactions on Neural Networks 3, 612-620 (1992)
12. Chentsov (Čencov), N.N.: Statistical Decision Rules and Optimal Inference. American Mathematical Society, Rhode Islandm U.S.A. (1982); originally published in Russian, Nauka, Moscow (1972)
13. Cichocki, A., Amari, S.: Adaptive Blind Signal and Image Processing. John Wiley, Chichester (2002)
14. Csiszár, I.: I-divergence geometry of probability distributions and minimization problems. The Annals of Probability 3, 146-158 (1975)
15. Csiszár, I., Tusnády, G.: Information geometry and alternating minimization procedures. In: Dedewicz, E.F., et al. (eds.) Statistics and Decisions, vol. (1), pp. 205-237. R. Oldenbourg Verlag, Munich (1984)
16. Ikeda, S., Tanaka, T., Amari, S.: Stochastic reasoning, free energy, and information geometry. Neural Computation 16, 1779-1810 (2004) (Pubitemid 38932656)
17. Lebanon, G.: Riemannian Geometry and Statistical Machine Learning. CMU-LTI-05-189, Carnegie-Mellon University (2005)
18. Lebanon, G., Lafferty, J.: Boosting and maximum likelihood for exponential models. Advances in Neural Information Processing Systems, vol. 14, pp. 447-451. MIT Press, Cambridge (2002)
19. Matsushima, Y.: The alpha EM algorithms: Surrogate likelihood maximization using alpha-logarithmic information measures. IEEE Transactions on Information Theory 49, 692-706 (2003)
20. Murata, N., Takenouchi, T., Kanamori, T., Eguchi, S.: Information geometry of Uboost and Bregman divergence. Neural Computation 16, 1437-1481 (2004) (Pubitemid 38740017)
21. Nakahara, H., Amari, S.: Information-geometric measure for neural spikes. Neural Computation 14, 2269-2316 (2002)
22. Nomizu, K., Sasaki, T.: Affine Differential Geometry. Cambridge University Press, Cambridge (1994)
23. Ohara, A.: Geodesics for dual connections and means on symmetric cone. Interal Equation and Operator Theory 50, 537-548 (2004)
24. Ohara, A., Tsuchiya, T.: An information geometric approach to polynomial-time interior-pint algorithms (submitted, 2008)
25. Rényi, A.: On measures of entropy and information. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 547-561. University of California Press (1961)
26. Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics 52, 479 (1988)
27. Wu, S., Amari, S.: Conformal transformation of kernel functions: A data-dependent way to improve support vector machine classifiers. Neural Processing Letters 15, 59-67 (2002) (Pubitemid 34149543)
28. Zhang, J.: Divergence function, duality, and convex analysis. Neural Computation 16, 159-195 (2004) (Pubitemid 37528419)