no. 3
Asymptotic quantization for probability measures on Riemannian manifolds
Iacobelli, Mikaela1, 2
1 University of Rome Sapienza, Department of Mathematics Guido Castelnuovo, Piazzale Aldo Moro 5, 00185 Rome, Italy.
2 Ecole Polytechnique, Centre de mathématiques Laurent Schwartz, 91128 Palaiseau cedex, France.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 770-785.

In this paper we study the quantization problem for probability measures on Riemannian manifolds. Under a suitable assumption on the growth at infinity of the measure we find asymptotic estimates for the quantization error, generalizing the results on R d . Our growth assumption depends on the curvature of the manifold and reduces, in the flat case, to a moment condition. We also build an example showing that our hypothesis is sharp.

Reçu le :
DOI : 10.1051/cocv/2015025
Classification : 49Q20
Mots clés : Quantization of measures, Riemannian manifolds
Iacobelli, Mikaela 1, 2

1 University of Rome Sapienza, Department of Mathematics Guido Castelnuovo, Piazzale Aldo Moro 5, 00185 Rome, Italy.
2 Ecole Polytechnique, Centre de mathématiques Laurent Schwartz, 91128 Palaiseau cedex, France. 
@article{COCV_2016__22_3_770_0,
     author = {Iacobelli, Mikaela},
     title = {Asymptotic quantization for probability measures on {Riemannian} manifolds},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {770--785},
     publisher = {EDP-Sciences},
     volume = {22},
     number = {3},
     year = {2016},
     doi = {10.1051/cocv/2015025},
     zbl = {1344.49074},
     mrnumber = {3527943},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2015025/}
}
TY  - JOUR
AU  - Iacobelli, Mikaela
TI  - Asymptotic quantization for probability measures on Riemannian manifolds
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2016
SP  - 770
EP  - 785
VL  - 22
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2015025/
DO  - 10.1051/cocv/2015025
LA  - en
ID  - COCV_2016__22_3_770_0
ER  - 
%0 Journal Article
%A Iacobelli, Mikaela
%T Asymptotic quantization for probability measures on Riemannian manifolds
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2016
%P 770-785
%V 22
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2015025/
%R 10.1051/cocv/2015025
%G en
%F COCV_2016__22_3_770_0
Iacobelli, Mikaela. Asymptotic quantization for probability measures on Riemannian manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 770-785. doi : 10.1051/cocv/2015025. http://www.numdam.org/articles/10.1051/cocv/2015025/

G. Bouchitté, C. Jimenez and M. Rajesh, Asymptotique d’un problème de positionnement optimal. C. R. Math. Acad. Sci. Paris 335 (2002) 853–858. | DOI | MR | Zbl

G. Bouchitté, C. Jimenez and M. Rajesh, Asymptotic analysis of a class of optimal location problems. J. Math. Pures Appl. 95 (2011) 382–419. | DOI | MR | Zbl

A. Brancolini, G. Buttazzo, F. Santambrogio and E. Stepanov, Long-term planning versus short-term planning in the asymptotical location problem. ESAIM: COCV 15 (2009) 509–524. | Numdam | MR | Zbl

J. Bucklew and G. Wise, Multidimensional Asymptotic Quantization Theory with rth Power Distortion Measures. IEEE Inform. Theory 28 (1982) 239–247. | DOI | MR | Zbl

S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Vol. 1730 of Lect. Notes Math. Springer-Verlag, Berlin Heidelberg (2000). | MR | Zbl

B. Kloeckner, Approximation by finitely supported measures. ESAIM: COCV 18 (2012) 343–359. | Numdam | MR | Zbl

S. Mosconi, P. Tilli, Γ-Convergence for the Irrigation Problem. J. Conv. Anal. 12 (2005) 145–158. | MR | Zbl

J.M. Lee, Riemannian manifolds. An introduction to curvature. Vol. 176 of Grad. Texts Math. Springer-Verlag, New York (1997). | MR | Zbl

J.G. Ratcliffe, Foundations of hyperbolic manifolds, 2nd edn. Vol. 149 of Grad. Texts Math. Springer, New York (2006). | MR | Zbl

C. Villani, Topics in Optimal Transportation, Vol. 58 of Grad. Studies Math. American Math. Soc., Providence RI (2003). | MR | Zbl

Cité par Sources :