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On the Rate of Convergence of Empirical Measures in ∞-transportation Distance

Published online by Cambridge University Press:  20 November 2018

Nicolás García Trillos  and
Dejan Slepčev
Nicolás García Trillos
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA. e-mail: [email protected], [email protected]
Dejan Slepčev
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA. e-mail: [email protected], [email protected]
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Abstract

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We consider random i.i.d. samples of absolutely continuous measures on bounded connected domains. We prove an upper bound on the $\infty $-transportation distance between the measure and the empirical measure of the sample. The bound is optimal in terms of scaling with the number of sample points.

Keywords

60B1060D0505C70optimal transportationoptimal matchinginfinity transportation distancemin-max distanceempirical measure
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 6 , 01 December 2015 , pp. 1358 - 1383
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Ajtai, M., Komlos, J., and Tusnády, G., On optimal matchings. Combinatorica 4(1984), no. 4, 259–264.http://dx.doi.Org/10.1007/BF02579135 Google Scholar
[2] Arora, S., Rao, S., and Vazirani, U., Expander flows, geometric embeddings anagraph partitioning. J. ACM (JACM) 56(2009), no. 2, Art. 5,1–37. Google Scholar
[3] Ball, J. M. and Zarnescu, A., Partial regularity and smooth topology-preserving approximations of rough domains. arxiv:1312.5156Google Scholar
[4] Berstein, S., On a modification of Chebyshev's inequality and of the error formula of Laplace. Ann.Sci. Inst. Sav., Sect. Math. 1(1924), no. 4, 38–49.Google Scholar
[5] Boissard, E., Simple bounds for convergence of empirical and occupation measures in 1-Wasserstein distance. Electron. J. Probab. 16(2011), no. 83, 2296–2333.http://dx.doi.org/10.1214/EJP.v16-958 Google Scholar
[6] Bolley, F., Guillin, A., and Villani, C., Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theory Related Fields 137(2007), no. 3-4, 541–593.http://dx.doi.org/10.1007/s00440-006-0004-7 Google Scholar
[7] Champion, T., Thierry, L., De Pascale, and Juutinen, P., The ∞-Wasserstein distance: local solutions and existence of optimal transport maps. SIAM J. Math. Anal. 40(2008), no. 1,1–20.http://dx.doi.org/10.1137/07069938X Google Scholar
[8] Chernoff, H., A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statistics 23(1952), 493–507.http://dx.doi.Org/10.1214/aoms/1177729330 Google Scholar
[9] Dobriaac, V. and Yukich, J. E., Asymptotics for transportation cost in high dimensions. J. Theoret. Probab. 8(1995), no. 1, 97–118.http://dx.doi.org/10.1007/BF02213456 Google Scholar
[10] Dudley, R. M., The speed of mean Glivenko-Cantelli convergence. Ann. Math. Statist. 40(1968), 40–50.http://dx.doi.Org/10.1214/aoms/1177697802 Google Scholar
[11] Leighton, T. and Shor, P., Tight bounds for minimax grid matching with applications to the average case analysis of algorithms. Combinatorica 9(1989), no. 2,161–187.http://dx.doi.org/10.1007/BF02124678 Google Scholar
[12] Shor, P. W. and Yukich, J. E., Minimax grid matching and empirical measures. Ann. Probab. 19(1991), no. 3, 1338–1348. http://dx.doi.Org/10.1214/aop/1176990347 Google Scholar
[13] Talagrand, M., The generic chaining. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.Google Scholar
[14] Talagrand, M., Upper and lower bounds of stochastic processes. A Series of Modern Surveys in Mathematics, 60, Springer-Verlag, Berlin Heidelberg, 2014 Google Scholar
[15] Talagrand, M., The transportation cost from the uniform measure to the empirical measure in dimension ≥ 3. Ann. Probab. 22(1994), no. 2, 919–959.http://dx.doi.Org/10.1214/aop/1176988735 Google Scholar
[16] Talagrand, M. and Yukich, J. E., The integrability of the square exponential transportation cost. Ann. Appl. Probab. 3(1993), no. 4, 1100–1111.http://dx.doi.org/10.1214/aoap/1177005274 Google Scholar