Medians and means in Finsler geometry

LMS Journal of Computation and Mathematics

Volumn 15, Issue , 2012, Pages 23-37

  Arnaudon, Marc ^a   Nielsen, Frank ^b,c  

^a UNIVERSITÉ DE POITIERS   (France)

^b ÉCOLE POLYTECHNIQUE   (France)

^c SONY CORPORATION   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84880853307     PISSN: 14611570     EISSN: None     Source Type: Journal    
DOI: 10.1112/S1461157010000513     Document Type: Article

References (37)

1. B. Afsari, 'Riemannian Lp center of mass: existence, uniqueness, and convexity', Proc. Amer. Math. Soc., S0002-9939(2010)10541-5, Article electronically published on 27 August 2010.
2. J. C. Alvarez Paiva, 'Some problems on Finsler geometry', Handbook of differential geometry II (Elsevier/North-Holland, Amsterdam, 2006) 1-33.
3. S. I. Amary and H. Nagaoka, Methods of information geometry, Translations of Mathematical Monographs 191 (American Mathematical Society and Oxford University Press, 2000).
4. M. Arnaudon and X. M. Li, 'Barycenters of measures transported by stochastic flows', Ann. Probab. 33 (2005) no. 4, 1509-1543.
5. L. Astola and L. Florack, 'Finsler geometry on higher order tensor fields and applications to high angular resolution diffusion imaging', Scale Space and Variational Methods in Computer Vision, Second International Conference, Voss, Norway, June 1-5, 2009, Int. J. Comput. Vis. 92 (2011) 325-336.
6. L. Auslander, 'On curvature in Finsler geometry', Trans. Amer. Math. Soc. 79 (1955) no. 2, 378-388.
7. D. Bao, S. S. Chern and Z. Shen, An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics (Springer, Berlin, 2000).
8. S. S. Chern and Z. Shen, Riemann-Finsler geometry, Nankai Tracts in Mathematics 6 (World Scientific, Singapore, 2005).
9. J. M. Corcuera and W. S. Kendall, 'Riemannian barycentres and geodesic convexity', Math. Proc. Cambridge Philos. Soc. 127 (1999) no. 2, 253-269.
10. E. Dzhafarov and H. Colonius, Fechnerian metrics in unidimensional and multidimensional stimulus spaces, Psychonomic Bulletin & Review 6 (Springer, New York, 1999) Issue 2.
11. C. Elkan, 'Using the triangle inequality to accelerate k-means', Proceedings of the Twentieth International Conference on Machine Learning (ICML), Washington, DC, 2003 (AAAI Press, Menlo Park, CA, 2003) 147-153.
12. M. Emery and G. Mokobodzki, 'Sur le barycentre d'une probabilité dans une variété', Séminaire de probabilités XXV, Lecture Notes in Mathematics 1485 (Springer, Berlin, 1991) 220-233.
13. P. T. Fletcher, S. Venkatasubramanian and S. Joshi, 'The geometric median on Riemannian manifolds with application to robust atlas estimation', NeuroImage 45 (2009) S143-S152.
14. B. Fuglede and F. Topsoe, 'Jensen-Shannon divergence and Hilbert space embedding', IEEE Int. Symp. Inf. Theory (2004) 31-31.
15. D. Fuhry, R. Jin and D. Zhang, 'Efficient skyline computation in metric space', Proceedings of the 12th International Conference on Extending Database Technology: Advances in Database Technology, EDBT'09 (ACM, New York, 2009) 1042-1051.
16. R. Gallego Torrome, 'On the generalization of theorems from Riemannian to Finsler geometry I: metric theorems', arXiv:math/0503704v3 [math.DG] 3 April 2008.
17. V. Garcia and F. Nielsen, 'Simplification and hierarchical representations of mixtures of exponential families', Signal Process. 90 (2010) no. 12, 3197-3212.
18. F. R. Hampel, P. J. Rousseeuw, E. M. Ronchetti and W. A. Stahel, Robust statistics the approach based on influence function (Wiley, New York, 1986).
19. H. Karcher, 'Riemannian center of mass and mollifier smoothing', Commun. Pure Appl. Math. XXX (1977) 509-541.
20. W. S. Kendall, 'Probability, convexity and harmonic maps with small image I: uniqueness and fine existence', Proc. Lond. Math. Soc. (3) 61 (1990) no. 2, 371-406.
21. W. S. Kendall, 'Convexity and the hemisphere', J. Lond. Math. Soc. (2) 43 (1991) no. 3, 567-576.
22. H. W. Kuhn, 'A note on Fermat's problem', Math. Program. 4 (1973) 98-107.
23. H. Le, 'Estimation of Riemannian barycentres', LMS J. Comput. Math. 7 (2004) 193-200.
24. J. Melonakos, E. Pichon, S. Angenent and A. Tannenbaum, 'Finsler active contours', IEEE Trans. Pattern Anal. Mach. Intell. 30 (2008) no. 3, 412-423.
25. F. Nielsen and R. Nock, 'Sided and symmetrized Bregman centroids', IEEE Trans. Inf. Theory 55 (2009) no. 66, 2882-2904.
26. S.-I. Ohta, 'Uniform convexity and smoothness, and their applications in Finsler geometry', Math. Ann. 343 (2009) 669-699.
27. L. M. Ostresh Jr., 'On the convergence of a class of iterative methods for solving the Weber location problem', Oper. Res. 26 (1978) no. 4.
28. M. Péchaud, R. Keriven and M. Descoteaux, 'Brain connectivity using geodesics in HARDI', IEEE International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI), London, September 2009, Lecture Notes in Computer Science 5762 (Springer, Berlin, 2009) 482-489.
29. B. Pelletier, 'Informative barycentres in statistics', Ann. Inst. Statist. Math. 57 (2005) no. 4, 767-780.
30. J. Picard, 'Barycentres et martingales dans les variétés', Ann. Inst. H. Poincaré Probab. Stat. 30 (1994) 647-702.
31. A. Sahib, 'Espérance d'une variable aléatoire a valeurs dans un espace métrique', Thèse de l'Université de Rouen, 1998.
32. Z. Shen, 'Riemann-Finsler geometry, with applications to information geometry', Chinese Ann. Math. Ser. B 27 (2006) 73-94.
33. Y. Vardi and C. H. Zhang, 'The multivariate L1-median and associated data depth', Proc. Natl. Acad. Sci. USA 97 (2000) 1423-1426.
34. E. Weiszfeld, 'Sur le point pour lequel la somme des distances de n points donnés est minimum', T̂ohoku Math. J. 43 (1937) 355-386.
35. B. Y. Wu and Y. L. Xin, 'Comparison theorems in Finsler geometry and their applications', Math. Ann. 337 (2007) 177-196.
36. L. Yang, 'Riemannian median and its estimation', LMS J. Comput. Math. 13 (2010) 461-479.
37. C. Zach, L. Shan and M. Niethammer, Globally optimal Finsler active contours, Lecture Notes in Computer Science 5748 (Springer, 2009) 552-561.