The dual Voronoi diagrams with respect to representational bregman divergences

6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009

Volumn , Issue , 2009, Pages 71-78

  Nielsen, Frank ^a,b   Nock, Richard ^c  

^a ÉCOLE POLYTECHNIQUE   (France)

^b SONY COMPUTER SCIENCE LABORATORIES INC   (Japan)

^c UNIVERSITÉ DES ANTILLES ET DE LA GUYANE   (France)

Author keywords

Bregman divergences; Centroids; F divergences; Power diagrams; Voronoi diagrams; divergences; divergences

Indexed keywords

COMPUTATIONAL GEOMETRY;

BREGMAN DIVERGENCES; CENTROIDS; CLOSED-FORM FORMULAE; DIFFERENTIABLE FUNCTIONS; INFORMATION GEOMETRY; POWER DIAGRAMS; STATISTICAL MANIFOLDS; VORONOI DIAGRAMS;

GRAPHIC METHODS;

EID: 77951472664     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/ISVD.2009.15     Document Type: Conference Paper

References (21)

1. J.-D. Boissonnat and M. Yvinec, Algorithmic Geometry. New York, NY, USA: Cambridge University Press, 1998.
2. G. Leibon and D. Letscher, "Delaunay triangulations and Voronoi diagrams for Riemannian manifolds," in SCG '00: Proceedings of the sixteenth annual symposium on Computational geometry. New York, NY, USA: ACM, 2000, pp. 341-349.
3. K. Onishi and H. Imai, "Voronoi diagram in statistical parametric space by Kullback-Leibler divergence," in SCG '97: Proceedings of the thirteenth annual symposium on Computational geometry. New York, NY, USA: ACM, 1997, pp. 463-465.
4. K. Onishi and H. Imai, "Voronoi diagram for the dually flat space by divergence," IPSJ SIG Notes, Tech. Rep. 42, 1997.
5. S. Amari and H. Nagaoka, Methods of Information Geometry, A. M. Society, Ed. Oxford University Press, 2000.
6. F. Nielsen, J.-D. Boissonnat, and R. Nock, "On Bregman Voronoi diagrams," in SODA '07: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, 2007, pp. 746-755.
7. F. Nielsen, J.-D. Boissonnat, and R. Nock, "Bregman Voronoi diagrams: Properties, algorithms and applications," CoRR, vol. abs/0709.2196, 2007.
8. L. M. Bregman, "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 Mathematical Physics, vol. 7, pp. 200-217, 1967.
9. 6: Advanced Rendering Techniques. Charles River Media, 2008.
10. S. Eguchi and J. Copas, "A class of logistic-type discriminant functions," Biometrika, vol. 89, no. 1, pp. 1-22, March 2002. [Online]. Available: http://dx.doi.org/10.1093/biomet/89.1.1
11. S.-i. Amari, "Integration of stochastic models by minimizing α-divergence," Neural Comput., vol. 19, no. 10, pp. 2780-2796, 2007.
12. P. Vos, "Geometry of f-divergence," Annals of the Institute of Statistical Mathematics, vol. 43, no. 3, pp. 515-537, September 1991. [Online]. Available: http://ideas.repec.org/a/spr/aistmt/v43y1991i3p515-537.html
13. I. Csiszár, "Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems," Ann. Stat., vol. 19, pp. 2032-2066, 1991.
14. M. Mihoko and S. Eguchi, "Robust blind source separation by beta divergence," Neural Comput., vol. 14, no. 8, pp. 1859-1886, 2002.
15. F. Nielsen and R. Nock, "Sided and symmetrized Bregman centroids," IEEE Transactions on Information Theory, 2009.
16. A. Banerjee, S. Merugu, I. S. Dhillon, and J. Ghosh, "Clustering with Bregman divergences," J. Mach. Learn. Res., vol. 6, pp. 1705-1749, 2005.
17. R. Nock, P. Luosto, and J. Kivinen, "Mixed Bregman clustering with approximation guarantees," in ECML PKDD '08: Proceedings of the European conference on Machine Learning and Knowledge Discovery in Databases. Berlin, Heidelberg: Springer-Verlag, 2008, pp. 154-169.
18. B. Chazelle, "An optimal convex hull algorithm in any fixed dimension," Discrete Computational & Geometry, vol. 10, pp. 377-409, 1993.
19. F. Aurenhammer, "Power diagrams: Properties, algorithms and applications," SIAM Journal of Computing, vol. 16, no. 1, pp. 78-96, 1987.
20. F. Nielsen and R. Nock, "On approximating the smallest enclosing bregman balls," in SCG '06: Proceedings of the twenty-second annual symposium on Computational geometry. New York, NY, USA: ACM, 2006, pp. 485-486.
21. F. Nielsen and R. Nock, "On the smallest enclosing information disk," Inf. Process. Lett., vol. 105, no. 3, pp. 93-97, 2008.