Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T11:55:28.076Z Has data issue: false hasContentIssue false
  • English
  • Français

Riemannian median and its estimation

Part of: Geometric probability and stochastic geometry Operations research and management science Calculus on manifolds; nonlinear operators

Published online by Cambridge University Press:  01 December 2010

Le Yang
Le Yang*
Affiliation:
Laboratoire de Mathématiques et, Applications (CNRS: UMR 6086) Université de Poitiers, Téléport 2 – BP30179, Boulevard Marie et Pierre Curie, F – 86962 Futuroscope Chasseneuil Cedex, France (email: [email protected])

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we define the geometric median for a probability measure on a Riemannian manifold, give its characterization and a natural condition to ensure its uniqueness. In order to compute the geometric median in practical cases, we also propose a subgradient algorithm and prove its convergence as well as estimating the error of approximation and the rate of convergence. The convergence property of this subgradient algorithm, which is a generalization of the classical Weiszfeld algorithm in Euclidean spaces to the context of Riemannian manifolds, also improves a recent result of P. T. Fletcher et al. [NeuroImage 45 (2009) S143–S152].

MSC classification

Secondary: 58C05: Real-valued functions 60D05: Geometric probability and stochastic geometry 90C25: Convex programming 90B85: Continuous location
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 461 - 479
Copyright
Copyright © London Mathematical Society 2010

References

[1] Alexandrov, A. D., ‘Über eine Verallgemeinerung der Riemannschen Geometrie’, Schr. Forschungsinst. Math. 1 (1957) 33–84.Google Scholar
[2] Aleksandrov, A. D. et al., ‘Generalized Riemannian spaces’, Russian Math. Surveys 41 (1986) no. 3, 1–54.CrossRefGoogle Scholar
[3] Arnaudon, M., ‘Espérances conditionnelles et C-martingales dans les variétés’, Séminaire de Probabilités-XXVIII, Lecture Notes in Mathematics 1583 (Springer, Berlin, 1994) 300–311.Google Scholar
[4] Arnaudon, M., ‘Barycentres convexes et approximations des martingales dans les variétés’, Séminaire de Probabilités-XXIX, Lecture Notes in Mathematics 1613 (Springer, Berlin, 1995) 70–85.CrossRefGoogle Scholar
[5] Arnaudon, M. and Li, X. M., ‘Barycenters of measures transported by stochastic flows’, Ann. Probab. 33 (2005) no. 4, 1509–1543.CrossRefGoogle Scholar
[6] Barbaresco, F., ‘Innovative tools for radar signal processing based on Cartan’s geometry of SPD matrices and information geometry’, IEEE International Radar Conference, May 26–30, Rome, Italy, 2008.CrossRefGoogle Scholar
[7] Barbaresco, F., ‘Interactions between symmetric cone and information geometries’, ETVC’08, Lecture Notes in Computer Science 5416 (Springer, Berlin, 2009) 124–163.Google Scholar
[8] Barbaresco, F., ‘New foundation of radar doppler signal processing based on advanced differential geometry of symmetric spaces: doppler matrix CFAR and radar application’, Radar’09 Conference, Bordeaux, October 2009.Google Scholar
[9] Barbaresco, F. and Bouyt, G., ‘Espace Riemannien symmetrique et géométrie des espaces de matrices de covariance : équations de diffusion et calculs de médianes’, XXII e Colloque GRETSI, Dijon, September 2009.Google Scholar
[10] Bridson, M. and Haefliger, A., Metric spaces of non-positive curvature (Springer, Berlin, 1999).Google Scholar
[11] Cheeger, J. and Ebin, D. G., Comparison theorems in Riemannian geometry (North Holland, Amsterdam, 1975).Google Scholar
[12] Correa, R. and Lemaréchal, C., ‘Convergence of some algorithms for convex minimization’, Math. Program. 62 (1993) 261–275.Google Scholar
[13] 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
[14] Ferreira, O. P. and Oliveira, P. R., ‘Subgradient algorithm on Riemannian manifolds’, J. Optim. Theory Appl. 97 (1998) no. 1, 93–104.CrossRefGoogle Scholar
[15] Fletcher, P. T. and Joshi, S., ‘Principle geodesic analysis on symmetric spaces: statistics of diffusion tensors’, Proceedings of ECCV Workshop on Computer Vision Approaches to Medical Image Analysis (2004), 87–98.Google Scholar
[16] Fletcher, P. T. et al., ‘Statistics of shape via principle geodesic analysis on Lie groups’, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2003) 95–101.Google Scholar
[17] Fletcher, P. T. et al., ‘The geometric median on Riemannian manifolds with application to robust atlas estimation’, NeuroImage 45 (2009) S143–S152.CrossRefGoogle ScholarPubMed
[18] Jost, J., Riemannian geometry and geometric analysis (Springer, Berlin, 2005).Google Scholar
[19] Karcher, H., ‘Riemannian center of mass and mollifier smoothing’, Commun. Pure Appl. Math. xxx (1977) 509–541.CrossRefGoogle Scholar
[20] Katz, I. N. and Cooper, L., ‘Optimal location on a sphere’, Comput Math. Appl. 6 (1980) 175–196.CrossRefGoogle Scholar
[21] Kendall, W. S., ‘Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence’, Proc. London Math. Soc. (3) 61 (1990) no. 2, 371–406.Google Scholar
[22] Kendall, W. S., ‘Convexity and the hemisphere’, J. London Math. Soc. (2) 43 (1991) no. 3, 567–576.CrossRefGoogle Scholar
[23] Kuhn, H. W., ‘A note on Fermat’s problem’, Math. Program. 4 (1973) 98–107, North-Holland Publishing Company.CrossRefGoogle Scholar
[24] Le, H., ‘Estimation of Riemannian barycentres’, LMS J. Comput. Math. 7 (2004) 193–200.CrossRefGoogle Scholar
[25] Nedic, A. and Bertsekas, D. P., ‘Convergence rate of incremental subgradient algorithms’, Stochastic optimization: algorithms and applications (eds Uryasev, S. and Pardalos, P. M.; Kluwer Academic Publishers, Dordrecht, 2000) 263–304.Google Scholar
[26] Nedic, A. and Bertsekas, D. P., ‘Incremental subgradient methods for non-differentiable optimization’, SIAM J. Optim. 12 (2001) no. 1, 109–138.Google Scholar
[27] Ostresh, L. M. Jr., ‘On the convergence of a class of iterative methods for solving Weber location problem’, Oper. Res. 26 (1978) no. 4, 597–609.Google Scholar
[28] Pennec, X., ‘Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements’, J. Math. Imaging Vision 25 (2006) 127–154.Google Scholar
[29] Picard, J., ‘Barycentres et martingales sur une variété’, Ann. Inst. H. Poincaré Probab. Statist 30 (1994) no. 4, 647–702.Google Scholar
[30] Sahib, A., ‘Espérance d’une variable aléatoire à valeur dans un espace métrique’, Thèse de l’Université de Rouen, 1998.Google Scholar
[31] Sakai, T., Riemannian geometry, Translations of Mathematical Monographs 149 (American Mathematical Society, Providence, RI, 1996).CrossRefGoogle Scholar
[32] Small, C. G., ‘Multidimensional medians arising from geodesics on graphs’, Ann. Statist. 25 (1997) no. 2, 478–494.CrossRefGoogle Scholar
[33] Udriste, C., Convex functions and optimization methods on Riemannian manifolds (Kluwer Academic Publishers, Dordrecht, 1994).Google Scholar
[34] Weiszfeld, E., ‘Sur le point pour lequel la somme des distances de n points donnés est minimum’, Tohoku Math. J. 43 (1937) 355–386.Google Scholar