# Means in complete manifolds: uniqueness and approximation

• Volume: 18, page 185-206
• ISSN: 1292-8100

top

## Abstract

top
Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure $\mu \left(x\right)=N{\sum }_{k=1}^{N}{x}_{k}$ μ ( x ) = 1 N ∑ k = 1 N δ x k has a uniquep–mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(ω)) is well-defined. Assume M is compact and consider a probability measure ν in M. Using partial simulated annealing, we define a continuous semimartingale which converges in probability to the set of minimizers of the integral of distance at power p with respect to ν. When the set is a singleton, it converges to the p–mean.

## How to cite

top

Arnaudon, Marc, and Miclo, Laurent. "Means in complete manifolds: uniqueness and approximation." ESAIM: Probability and Statistics 18 (2014): 185-206. <http://eudml.org/doc/273639>.

@article{Arnaudon2014,
abstract = {Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure $\mu (x)=N\sum _\{k=1\}^N\{x_k\}$ μ ( x ) = 1 N ∑ k = 1 N δ x k has a uniquep–mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(ω)) is well-defined. Assume M is compact and consider a probability measure ν in M. Using partial simulated annealing, we define a continuous semimartingale which converges in probability to the set of minimizers of the integral of distance at power p with respect to ν. When the set is a singleton, it converges to the p–mean.},
author = {Arnaudon, Marc, Miclo, Laurent},
journal = {ESAIM: Probability and Statistics},
keywords = {stochastic algorithms; diffusion processes; simulated annealing; homogenization; probability measures on compact riemannian manifolds; intrinsic p-means; instantaneous invariant measures; Gibbs measures; spectral gap at small temperature; probability measures on compact Riemannian manifolds; intrinsic $p$-means},
language = {eng},
pages = {185-206},
publisher = {EDP-Sciences},
title = {Means in complete manifolds: uniqueness and approximation},
url = {http://eudml.org/doc/273639},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Arnaudon, Marc
AU - Miclo, Laurent
TI - Means in complete manifolds: uniqueness and approximation
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 185
EP - 206
AB - Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure $\mu (x)=N\sum _{k=1}^N{x_k}$ μ ( x ) = 1 N ∑ k = 1 N δ x k has a uniquep–mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(ω)) is well-defined. Assume M is compact and consider a probability measure ν in M. Using partial simulated annealing, we define a continuous semimartingale which converges in probability to the set of minimizers of the integral of distance at power p with respect to ν. When the set is a singleton, it converges to the p–mean.
LA - eng
KW - stochastic algorithms; diffusion processes; simulated annealing; homogenization; probability measures on compact riemannian manifolds; intrinsic p-means; instantaneous invariant measures; Gibbs measures; spectral gap at small temperature; probability measures on compact Riemannian manifolds; intrinsic $p$-means
UR - http://eudml.org/doc/273639
ER -

## References

top
1. [1] B. Afsari, Riemannian Lp center of mass: existence, uniqueness, and convexity. Proc. Amer. Math. Soc. S 0002–9939 (2010) 10541-5. (electronic) Zbl1220.53040MR2736346
2. [2] B. Afsari, R. Tron and R. Vidal, On the convergence of gradient descent for finding the Riemannian center of mass. arXiv:1201.0925. Zbl1285.90031MR3057324
3. [3] M. Arnaudon and F. Nielsen, Medians and means in Finsler geometry. LMS J. Comput. Math.15 (2012) 23–37. Zbl1293.53025MR2891143
4. [4] M. Arnaudon, C. Dombry, A. Phan and L. Yang, Stochastic algorithms for computing means of probability measures Stoch. Proc. Appl.122 (2012) 1437–1455. Zbl1262.60073MR2914758
5. [5] M. Arnaudon and F. Nielsen, On computing the Riemannian 1-Center. Comput. Geom.46 (2013) 93–104. Zbl1259.65036MR2949613
6. [6] M. Bădoiu and K.L. Clarkson, Smaller core-sets for balls, Proc. of the fourteenth Annual ACM-SIAM Symposium on Discrete algorithms. Soc. Industrial Appl. Math. Philadelphia, PA, USA (2003) 801–802. Zbl1092.68660MR1974995
7. [7] R. Bhattacharya and V. Patrangenaru, Large sample theory of intrinsic and extrinsic sample means on manifolds (i). Ann. Statis.31 (2003) 1–29. Zbl1020.62026MR1962498
8. [8] S. Bonnabel, Convergence des méthodes de gradient stochastique sur les variétés riemanniennes. In GRETSI, Bordeaux (2011).
9. [9] H. Cardot, P. Cénac and P.-A. Zitt, Efficient and fast estimation of the geometric median in Hilbert spaces with an averaged stochastic gradient algorithm, Bernoulli. Zbl1259.62068
10. [10] B. Charlier, Necessary and sufficient condition for the existence of a Fréchet mean on the circle. arXiv:1109.1986. Zbl06282490MR3126155
11. [11] 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.
12. [12] D. Groisser, Newton’s method, zeroes of vector fields, and the Riemannian center of mass. Adv. Appl. Math.33 (2004) 95–135. Zbl1062.53025MR2064359
13. [13] D. Groisser, On the convergence of some Procrustean averaging algorithms. Stochastics77 (2005) 31–60. Zbl1079.60018MR2138772
14. [14] E.P. Hsu, Estimates of derivatives of the heat kernel on a compact Riemannian manifold. Proc. Amer. Math. Soc.127 (1999) 3739–3744. Zbl0948.58025MR1618694
15. [15] R. Holley, S. Kusuoka and D. Stroock, Asymptotics of the spectral gap with applications to the theory of simulated annealing. J. Funct. Anal.83 (1989) 333–347. Zbl0706.58075MR995752
16. [16] R. Holley and D. Stroock, Annealing via Sobolev inequalities. Commun. Math. Phys.115 (1988) 553–569. Zbl0643.60092MR933455
17. [17] T. Hotz and S. Huckemann, Intrinsic mean on the circle: Uniqueness, Locus and Asymptotics. arXiv:org1108:2141. Zbl1331.62269MR3297863
18. [18] W.S. Kendall, Probability, convexity and harmonic maps with small image I: uniqueness and fine existence. Proc. London Math. Soc.61 (1990) 371–406. Zbl0675.58042MR1063050
19. [19] H. Le, Estimation of Riemannian barycentres. LMS J. Comput. Math.7 (2004) 193–200. Zbl1054.60011MR2085875
20. [20] L. Miclo, Recuit simulé sans potentiel sur une variété compacte. Stoch. and Stochastic Reports41 (1992) 23–56. Zbl0758.60033MR1275365
21. [21] L. Miclo, Recuit simulé partiel, Stoch. Process. Appl.65 (1996) 281–298. Zbl0889.60073MR1425361
22. [22] S.J. Sheu, Some estimates of the transition density function of a nondegenerate diffusion Markov process. Ann. Probab.19 (1991) 538–561. Zbl0738.60060MR1106275
23. [23] K.T. Sturm, Probability measures on metric spaces of nonpositive curvature, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002). Contemp. Math. Amer. Math. Soc. 338 (2003) 357–390. Zbl1040.60002MR2039961
24. [24] E. Weiszfeld, Sur le point pour lequel la somme des distances de n points donnés est minimum. Tohoku Math. J.43 (1937) 355–386. Zbl0017.18007
25. [25] L. Yang, Riemannian median and its estimation. LMS J. Comput. Math.13 (2010) 461–479. Zbl1226.60018MR2748393
26. [26] L. Yang, Some properties of Frechet medians in Riemannian manifolds. Preprint.

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.