Citation Relationships



Zhang J (2004) Divergence function, duality, and convex analysis. Neural Comput 16:159-95 [PubMed]

References and models cited by this paper

References and models that cite this paper

Ackley DH, Hinton GE, Sejnowski TJ (1985) A learning algorithm for Bolzmann machines. Cognitive Sci 9:147-69

Amari S (1982) Differential geometry of curved exponential families-curvatures and information loss Ann Stat 10:357-385

Amari S (1991) Dualistic geometry of the manifold higher-order neurons Neural Netw 4:443-451

Amari S (1995) Information geometry of EM and EM algorithms for neural networks Neural Netw 8:1379-1408

Amari S, Ikeda S, Shimokawa H (2001) Information geometry and meanfield approximation: The Œ+ projection approach Advanced mean field methods: Theory and practive, Opper M:Saad D, ed. pp.241

Amari S, Kurata K, Nagaoka H (1992) Information geometry of Boltzmann machines. IEEE Trans Neural Netw 3:260-71 [Journal] [PubMed]

Amari S, Nagaoka H (2000) Methods of information geometry

Amari SI (1985) Differential-geometrical methods in statistics

Bauschke HH, Borwein JM, Combettes PL (2002) Bregman monotone optimization algorithms CECM Preprint 02:184 (Available on-line: http:--www.cecm.sfu.ca-preprints-2002pp.html)

Bauschke HH, Combettes PL (2002) Iterating Bregman retractions CECM Preprint 02:186 (Available on-line: http:--www.cecm.sfu.ca-preprints-2002pp.html)

Bregman LM (1967) 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

Chentsov NN (1982) Statistical decision rules and optimal inference

Csiszar I (1967) On topical properties of f-divergence Studia Mathematicarum Hungarica 2:329-339

della_Pietra S, dDella_Pietra V, Lafferty J (2002) Duality and auxiliary functions for Bregman distances Tech. Rep. No. CMU-CS-01-109

Eguchi S (1983) Second order efficiency of minimum contrast estimators in a curved exponential family Ann Stat 11:793-803

Eguchi S (1992) Geometry of minimum contrast Hiroshima Mathematical Journal 22:631-647

Eguchi S (2002) U-boosting method for classification and information geometry Paper presented at the SRCCS International Statistical Workshop

Hardy G, Littlewood JE, Polya G (1952) Inequalities

Ikeda S, Amari S, Nakahara H (1999) Convergence of the wake-sleep algorithm Advances in neural information processing systems, Kearns M:Solla S:Cohon D, ed. pp.239

Kaas RE, Vos PW (1997) Geometric foundation of asymptotic inference

Kurose T (1994) On the divergences of 1-conformally flat statistical manifolds Tohoko Mathematical Journal 46:427-433

Lafferty J, della_Pietra S, della_Pietra V (1997) Statistical learning algorithms based on Bregman distances Proceedings of 1997 Canadian Workshop on Information Theory :77-80

Lauritzen S (1987) Statistical manifolds Differential geometry in statistical inference, Amari S:Barndorff-Nielsen O:Kass R:Lauritzen S:Rao CR, ed. pp.163

Lebanon G, Lafferty J (2002) Boosting and maximum likelihood for exponential models Advances in neural information processing systems, Dietterich TG:Becker S:Ghahramani Z, ed. pp.447

Matsuzoe H (1998) On realization of conformally-projectively flat statistical manifolds and the divergences Hokkaido Mathematical Journal 27:409-421

Matsuzoe H (1999) Geometry of contrast functions and conformal geometry Hiroshima Mathematical Journal 29:175-191

Matumoto T (1993) Any statistical manifold has a contrast function-On the C3-functions taking the minimum at the diagonal of the product manifold Hiroshima Mathematical Journal 23:327-332

Mihoko M, Eguchi S (2002) Robust blind source separation by beta divergence. Neural Comput 14:1859-86 [Journal] [PubMed]

Rao CR (1987) Differential metrics in probability spaces Differential geometry in statistical inference, Amari S:Barndorff-Nielson O:Kass R:Lauritzen S:Rao CR, ed. pp.217

Rockafellar RT (1970) Convex analysis

Shima H (1978) Compact locally Hessian manifolds Osaka Journal Of Mathematics 15:509-513

Shima H, Yagi K (1997) Geometry of Hessian manifolds Differential Geometry And Its Applications 7:277-290

Takeuchi J, Amari S (2004) Parallel prior and its properties IEEE Transaction on Information Theory (submitted)

Uohashi K, Ohara A, Fujii T (2000) 1-Conformally flat statistical submanifolds Osaka Journal Of Mathematics 37:501-507

Zhu HY, Rohwer R (1995) Bayesian invariant measurements of generalization Neural Processing Letter 2:28-31

Zhu HY, Rohwer R (1997) Measurements of generalisation based on information geometry Mathematics of neural networks: Model algorithms and applications, Ellacott SW:Mason JC:Anderson IJ, ed. pp.394

Amari S (2007) Integration of stochastic models by minimizing alpha-divergence. Neural Comput 19:2780-96 [Journal] [PubMed]

(37 refs)