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Approximation by finitely supported measures

Published online by Cambridge University Press:  13 April 2011

Benoît Kloeckner
Benoît Kloeckner*
Affiliation:
Institut Fourier, Université Joseph Fourier, BP 53, 38041 Grenoble, France. [email protected]
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Abstract

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.

Keywords

MeasuresWasserstein distancequantizationlocation problemcentroidal Voronoi tessellations
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 2 , April 2012 , pp. 343 - 359
Copyright
© EDP Sciences, SMAI, 2011

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References

Bouchitté, G., Jimenez, C. and Rajesh, M., Asymptotique d’un problème de positionnement optimal. C. R. Math. Acad. Sci. Paris 335 (2002) 853–858. Google Scholar
Brancolini, A., Buttazzo, G., Santambrogio, F. and Stepanov, E., Long-term planning versus short-term planning in the asymptotical location problem. ESAIM : COCV 15 (2009) 509–524. Google Scholar
Champion, T., De Pascale, L. and Juutinen, P., The ∞-Wasserstein distance : local solutions and existence of optimal transport maps. SIAM J. Math. Anal. 40 (2008) 1–20. Google Scholar
Dobrić, V. and Yukich, J.E., Asymptotics for transportation cost in high dimensions. J. Theoret. Probab. 8 (1995) 97–118. Google Scholar
Du, Q. and Wang, D., The optimal centroidal Voronoi tessellations and the Gersho’s conjecture in the three-dimensional space. Comput. Math. Appl. 49 (2005) 1355–1373. Google Scholar
Du, Q., Faber, V. and Gunzburger, M., Centroidal Voronoi tessellations : applications and algorithms. SIAM Rev. 41 (1999) 637–676. Google Scholar
K.J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics 85. Cambridge University Press (1986).
Fejes Tóth, L., Sur la représentation d’une population infinie par un nombre fini d’éléments. Acta. Math. Acad. Sci. Hungar 10 (1959) 299–304. Google Scholar
L. Fejes Tóth, 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).
S. Graf and H. Luschgy, Foundations of quantization for probability distributions, Lecture Notes in Mathematics 1730. Springer-Verlag (2000).
J. Heinonen, Lectures on analysis on metric spaces. Universitext, Springer-Verlag (2001).
Horowitz, J. and Karandikar, R.L., Mean rates of convergence of empirical measures in the Wasserstein metric. J. Comput. Appl. Math. 55 (1994) 261–273. Google Scholar
Hutchinson, J.E., Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981) 713–747. Google Scholar
Morgan, F. and Bolton, R., Hexagonal economic regions solve the location problem. Amer. Math. Monthly 109 (2002) 165–172. Google Scholar
Mosconi, S.J.N. and Tilli, P., Γ-convergence for the irrigation problem. J. Convex Anal. 12 (2005) 145–158. Google Scholar
Newman, D.J., The hexagon theorem. IEEE Trans. Inform. Theory 28 (1982) 137–139. Google Scholar
C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics 58. American Mathematical Society (2003).