Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T04:54:51.674Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform decomposition of probability measures: quantization, clustering and rate of convergence

Part of: Distribution theory - Probability Probability theory on algebraic and topological structures Distribution theory Limit theorems

Published online by Cambridge University Press:  16 January 2019

Julien Chevallier
Julien Chevallier*
Affiliation:
Université Grenoble Alpes
*
* Postal address: Université Grenoble Alpes, CNRS, LJK, 38000 Grenoble, France. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The study of finite approximations of probability measures has a long history. In Xu and Berger (2017), the authors focused on constrained finite approximations and, in particular, uniform ones in dimension d=1. In the present paper we give an elementary construction of a uniform decomposition of probability measures in dimension d≥1. We then use this decomposition to obtain upper bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained by Xu and Berger (2017) and to be sharp for generic probability measures.

Keywords

Uniform approximationWasserstein distancerate of convergencequantizationclustering

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62E17: Approximations to distributions (nonasymptotic) 60B10: Convergence of probability measures 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1037 - 1045
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bennett, W. R. (1948). Spectra of quantized signals. Bell Systems Tech. J. 27, 446–472.Google Scholar
[2]Boissard, E. and Le Gouic, T. (2014). On the mean speed of convergence of empirical and occupation measures in Wasserstein distance. Ann. Inst. H. Poincaré Prob. Statist. 50, 539–563.Google Scholar
[3]Bolley, F., Guillin, A. and Villani, C. (2007). Quantitative concentration inequalities for empirical measures on non-compact spaces. Prob. Theory Relat. Fields 137, 541–593.Google Scholar
[4]Bradley, P., Bennett, K. and Demiriz, A. (2000). Constrained k-means clustering. Tech. Rep. MSR-2000-65, Microsoft Research.Google Scholar
[5]Chen, Y., Welling, M. and Smola, A. (2012). Super-samples from kernel herding. Preprint. Available at http://arxiv.org/abs/1203.3472.Google Scholar
[6]Chevallier, J., Duarte, A., Löcherbach, E. and Ost, G. (2019). Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels. Stoch. Process. Appl. 124, 1–27.Google Scholar
[7]Dick, J., Kuo, F. Y. and Sloan, I. H. (2013). High-dimensional integration: the quasi-Monte Carlo way. Acta Numer. 22, 133–288.Google Scholar
[8]Faundez-Zanuy, M. and Pascual-Gaspar, J. M. (2011). Efficient on-line signature recognition based on multi-section vector quantization. Pattern Anal. Appl. 14, 37–45.Google Scholar
[9]Fournier, N. and Guillin, A. (2015). On the rate of convergence in Wasserstein distance of the empirical measure. Prob. Theory Relat. Fields 162, 707–734.Google Scholar
[10]Giles, M. B., Hefter, M., Mayer, L.and Ritter, K. (2018). Random bit quadrature and approximation of distributions on Hilbert spaces. Found. Comput. Math. Available at https://doi.org/10.1007/s10208-018-9382-3.Google Scholar
[11]Graf, S.and Luschgy, H. (2000). Foundations of Quantization for Probability Distributions. Springer, Berlin.Google Scholar
[12]Kloeckner, B. (2012). Approximation by finitely supported measures. ESAIM Control Optim. Calc. Var. 18, 343–359.Google Scholar
[13]Mak, S. and Joseph, V. R. (2018). Support points. Ann. Statist. 46, 2562–2592.Google Scholar
[14]Malinen, M. I. and Fränti, P. (2014). Balanced K-means for clustering. In Structural, Syntactic, and Pattern Recognition, (Lecture Notes Comput. Sci. 8621), Springer, Berlin, pp. 32–41.Google Scholar
[15]Pagès, G. (2015). Introduction to vector quantization and its applications for numerics. ESAIM Proc. Surveys 48, 29–79.Google Scholar
[16]Pagès, G. and Wilbertz, B.(2012). Optimal Delaunay and Voronoi quantization methods for pricing American style options. In Numerical methods in Finance, Springer, (Springer Proc. Math. 12), Berlin, pp. 171–217.Google Scholar
[17]Xu, C. and Berger, A. (2017). Best finite constrained approximations of one-dimensional probabilities. Preprint. Available at http://arxiv.org/abs/1704.07871.Google Scholar
[18]Zador, P. (1963). Development and evaluation of procedures for quantizing multivariate distributions. Tech. Rep. Stanford University.Google Scholar
[19]Zador, P. (1982). Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory 28, 139–149.Google Scholar