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Convergence rates for estimators of geodesic distances and Frèchet expectations

Part of: Statistics Nonparametric inference Multivariate analysis

Published online by Cambridge University Press:  16 January 2019

Catherine Aaron  and
Olivier Bodart
Catherine Aaron*
Affiliation:
Université Clermont Auvergne
Olivier Bodart*
Affiliation:
Université Jean Monnet
*
* Postal address: Laboratoire de Mathématiques Blaise Pascal (UMR 6620 - CNRS), Université Clermont Auvergne (Clermont-Ferrand 2), 63177 Aubière cedex, France. Email address: catherine.aaron@uca.fr
** Postal address: Institut Camille Jordan Faculté des Sciences et Techniques, Université Jean Monnet,42023 Saint-Étienne Cedex 2, France.
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Abstract

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Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.

Keywords

Geometric inferencegeodesic distancestatistics on manifoldsFréchet expectations

MSC classification

Primary: 62-07: Data analysis
Secondary: 62G05: Estimation 62G20: Asymptotic properties 62H99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1001 - 1013
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Baillo, A., Cuevas, A. and Justel, A. (2000). Set estimation and nonparametric detection. Canad. J. Statist. 28, 765–782.Google Scholar
[2]Brito, M. R., Quiroz, A. J. and Yukich, J. E. (2013). Intrinsic dimension identification via graph-theoretic methods. J. Multivariate Anal. 116, 263–277.Google Scholar
[3]Cormen, T. H., Leiserson, C. E., Rivest, R. L. and Stein, C. (2001). Introduction to Algorithms, 3rd edn. MIT Press, Cambridge, MA.Google Scholar
[4]Cuevas, A. and Rodríguez-Casal, A. (2004). On boundary estimation. Adv. Appl. Prob. 36, 340–354.Google Scholar
[5]Demartines, P. and Herault, J. (1997). Curvilinear component analysis: a self-organizing neural network for nonlinear mapping of data sets. IEEE Trans. Neural Networks 8, 148–154.Google Scholar
[6]Granata, D. and Carnevale, V. (2016). Accurate estimation of the intrinsic dimension using graph distances: unraveling the geometric complexity of datasets. Sci. Rep. 6, 31377.Google Scholar
[7]Howard, C. D. and Newman, C. M. (2001). Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Prob. 29, 577–623.Google Scholar
[8]Hwang, S. J., Damelin, S. B. and Hero III, A. O. (2016). Shortest path through random points. Ann. Appl. Prob. 26, 2791–2823.Google Scholar
[9]Kruskal, J. B. (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29, 1–27.Google Scholar
[10]Lee, J. A., Lendasse, A. and Verleyzen, M. (2004). Nonlinear projection with curvilinear distances: isomap versus curvilinear distance analysis. Neurocomputing 57, 49–76.Google Scholar
[11]Lennon, M., Mercier, G., Mouchot, M. C. and Hubert-Moy, L. (2001). Curvilinear component analysis for nonlinear dimensionality reduction of hyperspectral images. Proc. SPIE Image and Signal Process. for Remote Sens. VII, 4541, 157–168.Google Scholar
[12]Nilsson, J., Fioretos, T., Höglund, M. and Fontes, M. (2004). Approximate geodesic distances reveal biologically relevant structures in microarray data. Bioinformatics 20, 874–880.Google Scholar
[13]Pennec, X. (2006). Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vision 25, 127–154.Google Scholar
[14]Penrose, M. D. (1997). The longest edge of the random minimal spanning tree. Ann. Appl. Prob. 7, 340–361.Google Scholar
[15]Penrose, M. D. (1999). A strong law for the largest nearest-neighbour link between random points. J. London Math. Soc 60, 951–960.Google Scholar
[16]Saul, L. K. and Roweis, S. T. (2003). Think globally, fit locally: unsupervised learning of low dimensional manifolds. J. Mach. Learn. Res. 4, 119–155.Google Scholar
[17]Takens, F. (1985). On the numerical determination of the dimension of an attractor. In Dynamical Systems and Bifurcations, (Lectures Notes Math. 1125), Springer, Berlin, pp. 99–106.Google Scholar
[18]Tenenbaum, J. B., de Silva, V. and Langford, J. C. (2000). A global geometric framework for nonlinear dimensionality a global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323.Google Scholar