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Estimation of Riemannian Barycentres

Published online by Cambridge University Press:  01 February 2010

Huiling Le
Huiling Le
Affiliation:
School of Mathematical Sciences,University of Nottingham, University Park, Nottingham, NG7 2RDUnited Kingdom, [email protected], http://www.maths.nottingham.ac.uk/htbin-local/staff.info?hl

Abstract

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Using Jacobi field arguments, this paper describes an iterative procedure for finding the Riemannian barycentres of a class of probability measures on complete, simply connected Riemannian manifolds with a finite upper bound on their sectional curvatures. This, in particular, generalises an earlier result of the author's (‘Locating Fréchet means with application to shape spaces’, Adv. Appl. Probab. 33 (2001) 324-338).

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 7 , 2004 , pp. 193 - 200
Copyright
Copyright © London Mathematical Society 2004

References

1. Cheeger, J. and Ebin, D. G. , Comparison theorems in Riemannian geometry (North Holland, Amsterdam, 1975).Google Scholar
2. Corcuera, J. M. and Kendall, W. S., ‘Riemannian barycentres and geodesic convexity’, Math. Proc. Camb. Phil. Soc. 127 (1999) 253–269.CrossRefGoogle Scholar
3. Émery, M. and Mokobodzki, G., ‘Sur Le barycentre d'une Probabilite dans une variéte’, Séminaire Probabilite XXV, Lect. Notes in Math. 1485 (Springer, New York, 1991) 220–233.CrossRefGoogle Scholar
4. Jost, J., ‘Equilibrium maps between metric spaces’, Calc. Var. 2 (1994) 173–204.CrossRefGoogle Scholar
5. Jost, J., Riemannian geometry and geometric analysis (Springer, Berlin, 1998).CrossRefGoogle Scholar
6. Karcher, H., ‘Riemannian center of mass and mollifier smoothing’, Comm. Pure Appl. Math. 30(1977)509–541.CrossRefGoogle Scholar
7. Kendall, W. S., ‘Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence’, Proc. London Math. Soc. 61 (1990) 371–406.CrossRefGoogle Scholar
8. Kendall, W. S. ‘Convexity and the hemisphere’, J. London Math. Soc. 43 (1991) 567–576.CrossRefGoogle Scholar
9. Kume, A. and Le, H., ‘On Fréchet means in simplex shape spaces’, Adv. Appl. Prob. 35 (2003) 885–897.CrossRefGoogle Scholar
10. Le, H., ‘Mean size-and-shapes and mean shapes: a geometric point of view’, Adv. Appl. Probab. 27 (1995) 44–55.CrossRefGoogle Scholar
11. Le, H., ‘Locating Fréchet means with application to shape spaces’, Adv. Appl. Probab. 33 (2001) 324–338.CrossRefGoogle Scholar
12. Oller, J. M. and Corcuera, J. M., ‘Intrinsic analysis of the statistical estimation’, Ann. Statist. 23 (1995) 1562–1581.CrossRefGoogle Scholar