Jensen-Bregman Voronoi diagrams and centroidal tessellations

ISVD 2010 - 7th International Symposium on Voronoi Diagrams in Science and Engineering

Volumn , Issue , 2010, Pages 56-65

  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

divergence; Bhattacharyya distance; Bregman divergences; Burbea Rao divergence; Exponential families; F divergence; Hellinger Kakutani Matusita divergences; Information geometry; Jensen difference; Jensen's inequality; Jensen Shannon divergence

Indexed keywords

BHATTACHARYYA DISTANCE; BREGMAN DIVERGENCES; BURBEA-RAO DIVERGENCE; EXPONENTIAL FAMILY; F-DIVERGENCE; HELLINGER-KAKUTANI- MATUSITA DIVERGENCES; INFORMATION GEOMETRY; JENSEN DIFFERENCE; JENSEN'S INEQUALITY; JENSEN-SHANNON DIVERGENCE;

COMPUTATIONAL GEOMETRY; ENCODING (SYMBOLS); EXPONENTIAL FUNCTIONS; FUNCTION GENERATORS; IMAGE SEGMENTATION; INFORMATION THEORY; POISSON DISTRIBUTION;

GRAPHIC METHODS;

EID: 77955693545     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/ISVD.2010.17     Document Type: Conference Paper

References (28)

1. A. Okabe, B. Boots, K. Sugihara, and S. N. Chiu, Spatial tessellations: Concepts and applications of Voronoi diagrams, 2nd ed., ser. Probability and Statistics. NYC: Wiley, 2000, 671 pages.
2. J. Lin, "Divergence measures based on the Shannon entropy," IEEE Transactions on Information Theory, vol.37, pp. 145 - 151, 1991.
3. J.-D. Boissonnat, F. Nielsen, and R. Nock, "Bregman Voronoi diagrams," Discrete & Computational Geometry, 2010, accepted, extend ACM-SIAM SODA 2007.
4. T. M. Cover and J. A. Thomas, Elements of information theory. New York, NY, USA: Wiley-Interscience, 1991.
5. H. Jeffreys, "An invariant form for the prior probability in estimation problems," Proceedings ofthe Royal Society of London, vol.186, no.1007, pp. 453 - 461, March 1946.
6. M. D. Reid and R. C. Williamson, "Generalised Pinsker inequalities," CoRR, vol. abs/0906.1244, 2009, published at COLT'09.
7. P. Chen, Y. Chen, and M. Rao, "Metrics defined by Bregman divergences: Part I," Commun. Math. Sci., vol.6, pp. 9915-9926, 2008.
8. -, "Metrics defined by Bregman divergences: Part II," Commun. Math. Sci., vol.6, pp. 927 - 948, 2008.
9. J. Burbea and C. R. Rao, "On the convexity of some divergence measures based on entropy functions," IEEE Transactions on Information Theory, vol.28, no.3, pp. 489 - 495, 1982. (Pubitemid 12524012)
10. A. Banerjee, S. Merugu, I. S. Dhillon, and J. Ghosh, "Clustering with Bregman divergences," J. Mach. Learn. Res.,vol.6, pp. 1705 - 1749, 2005.
11. F. Nielsen and R. Nock, "Sided and symmetrized Bregman centroids," IEEE Transactions on Information Theory, vol.55, no.6, pp. 2048 - 2059, June 2009.
12. F. Nielsen and V. Garcia, "Statistical exponential families: A digest with flash cards," 2009, arXiv.org:0911.4863.
13. A. Bhattacharyya, "On a measure of divergence between two statistical populations defined by their probability distributions," Bulletin of Calcutta Mathematical Society, vol.35, pp. 99 - 110, 1943.
14. K. Matusita, "Decision rules based on the distance, for problems of fit, two samples, and estimation," Annal of Mathematics and Statistics, vol.26, pp. 631 - 640, 1955.
15. V. S. Huzurbazar, "Exact forms of some invariants for distributions admitting sufficient statistics," Biometrika, vol.42, no.3-4, pp. 533 - 573, 1955.
16. T. Kailath, "The divergence and Bhattacharyya distance measures in signal selection," Communications, IEEE Transactions on [legacy, pre - 1988], vol.15, no.1, pp. 52 - 60, 1967.
17. T. Jebara and R. Kondor, "Bhattacharyya and expected likelihood kernels," in 16th Annual Conference on Learning Theory and 7th Kernel Workshop, COLT/Kernel, 2003, p. 57.
18. C. Icking and L. Ha, "A tight bound for the complexity of Voronoi diagrams under polyhedral convex distance functions in 3d," in STOC '01: Proceedings of the thirty-third annual ACM symposium on Theory of computing. New York, NY, USA: ACM, 2001, pp. 316 - 321.
19. L. Ma, "Bisectors and Voronoi diagrams for convex distance functions," 2000, PhD thesis.
20. M. Dickerson, D. Eppstein, and K. A. Wortman, "Dilation, smoothed distance, and minimization diagrams of convex functions," arXiv 0812.0607.
21. M. I. Karavelas and M. Yvinec, "The Voronoi diagram of planar convex objects," in European Symposium on Algorithms (ESA), 2003, pp. 337 - 348.
22. F. Nielsen and R. Nock, "Jensen-Bregman divergences: Geometry and algorithms," 2010, research memo.
23. M. Balzer, T. Schlomer, and O. Deussen, "Capacity-constrained point distributions: a variant of Lloyd's method," ACM Trans. Graph., vol.28, no.3, 2009.
24. A. Yuille and A. Rangarajan, "The concave-convex procedure," Neural Computation, vol.15, no.4, pp. 915 - 936, 2003.
25. F. Nielsen and R. Nock, "The dual Voronoi diagrams with respect to representational bregman divergences," in International Symposium on Voronoi Diagrams (ISVD).DTU Lyngby, Denmark: IEEE, June 2009.
26. J. Burbea and C. R. Rao, "On the convexity of higher order Jensen differences based on entropy functions," IEEE Transactions on Information Theory, vol.28, no.6, pp. 961-1982.
27. P. Vos, "Geometry of f-divergence," Annals of the Institute of Statistical Mathematics, vol.43, no.3, pp. 515 - 537, 1991.
28. F. Nielsen and S. Boltz, "The Burbea-Rao and Bhattacharyya centroids," Computing Research Repository (CoRR),vol. http://arxiv.org/, April 2010.