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Medians and means in Finsler geometry

Part of: Local differential geometry

Published online by Cambridge University Press:  01 February 2012

Marc Arnaudon  and
Frank Nielsen
Marc Arnaudon
Affiliation:
Laboratoire de Mathématiques et Applications, CNRS: UMR 6086, Université de Poitiers, Téléport 2 – BP 30179, F-86962 Futuroscope, Chasseneuil Cedex, France (email: marc.arnaudon@math.univ-poitiers.fr)
Frank Nielsen
Affiliation:
Laboratoire d’Informatique (LIX), École Polytechnique, 91128 Palaiseau Cedex, France Sony Computer Science Laboratories, Inc, Tokyo, Japan (email: frank.nielsen@acm.org)

Abstract

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We investigate existence and uniqueness of p-means ep and the median e1 of a probability measure μ on a Finsler manifold, in relation with the convexity of the support of μ. We prove that ep is the limit point of a continuous time gradient flow. Under some additional condition which is always satisfied for p≥2, a discretization of this path converges to ep. This provides an algorithm for determining the Finsler center points.

MSC classification

Secondary: 53B40: Finsler spaces and generalizations (areal metrics)
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 23 - 37
Copyright
Copyright © London Mathematical Society 2012

References

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