We consider the problem of approximating a probability measure defined on a metric space by a measure supported on a finite number of points. More specifically we seek the asymptotic behavior of the minimal Wasserstein distance to an approximation when the number of points goes to infinity. The main result gives an equivalent when the space is a Riemannian manifold and the approximated measure is absolutely continuous and compactly supported.
Mots clés : measures, Wasserstein distance, quantization, location problem, centroidal Voronoi tessellations
@article{COCV_2012__18_2_343_0,
author = {Kloeckner, Beno{\^\i}t},
title = {Approximation by finitely supported measures},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {343--359},
publisher = {EDP-Sciences},
volume = {18},
number = {2},
year = {2012},
doi = {10.1051/cocv/2010100},
mrnumber = {2954629},
zbl = {1246.49040},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2010100/}
}
TY - JOUR AU - Kloeckner, Benoît TI - Approximation by finitely supported measures JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 343 EP - 359 VL - 18 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010100/ DO - 10.1051/cocv/2010100 LA - en ID - COCV_2012__18_2_343_0 ER -
%0 Journal Article %A Kloeckner, Benoît %T Approximation by finitely supported measures %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 343-359 %V 18 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010100/ %R 10.1051/cocv/2010100 %G en %F COCV_2012__18_2_343_0
Kloeckner, Benoît. Approximation by finitely supported measures. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 343-359. doi : 10.1051/cocv/2010100. http://www.numdam.org/articles/10.1051/cocv/2010100/
[1] , and , Asymptotique d'un problème de positionnement optimal. C. R. Math. Acad. Sci. Paris 335 (2002) 853-858. | MR | Zbl
[2] , , and , Long-term planning versus short-term planning in the asymptotical location problem. ESAIM : COCV 15 (2009) 509-524. | Numdam | MR | Zbl
[3] , and , The ∞-Wasserstein distance : local solutions and existence of optimal transport maps. SIAM J. Math. Anal. 40 (2008) 1-20. | MR | Zbl
[4] and , Asymptotics for transportation cost in high dimensions. J. Theoret. Probab. 8 (1995) 97-118. | MR | Zbl
[5] and , The optimal centroidal Voronoi tessellations and the Gersho's conjecture in the three-dimensional space. Comput. Math. Appl. 49 (2005) 1355-1373. | MR | Zbl
[6] , and , Centroidal Voronoi tessellations : applications and algorithms. SIAM Rev. 41 (1999) 637-676. | MR | Zbl
[7] , The geometry of fractal sets, Cambridge Tracts in Mathematics 85. Cambridge University Press (1986). | MR | Zbl
[8] , Sur la représentation d'une population infinie par un nombre fini d'éléments. Acta. Math. Acad. Sci. Hungar 10 (1959) 299-304. | MR | Zbl
[9] , Lagerungen in der Ebene, auf der Kugel und im Raum, Die Grundlehren der mathematischen Wissenschaften, Band 65. Zweite verbesserte und erweiterte Auflage, Springer-Verlag (1972). | MR | Zbl
[10] and , Foundations of quantization for probability distributions, Lecture Notes in Mathematics 1730. Springer-Verlag (2000). | MR | Zbl
[11] , Lectures on analysis on metric spaces. Universitext, Springer-Verlag (2001). | MR | Zbl
[12] and , Mean rates of convergence of empirical measures in the Wasserstein metric. J. Comput. Appl. Math. 55 (1994) 261-273. | MR | Zbl
[13] , Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981) 713-747. | MR | Zbl
[14] and , Hexagonal economic regions solve the location problem. Amer. Math. Monthly 109 (2002) 165-172. | MR | Zbl
[15] and , Γ-convergence for the irrigation problem. J. Convex Anal. 12 (2005) 145-158. | MR | Zbl
[16] , The hexagon theorem. IEEE Trans. Inform. Theory 28 (1982) 137-139. | MR | Zbl
[17] , Topics in optimal transportation, Graduate Studies in Mathematics 58. American Mathematical Society (2003). | MR | Zbl
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