Electronic Journal of Statistics

Rates of convergence for robust geometric inference

Frédéric Chazal, Pascal Massart, and Bertrand Michel

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Distances to compact sets are widely used in the field of Topological Data Analysis for inferring geometric and topological features from point clouds. In this context, the distance to a probability measure (DTM) has been introduced by Chazal et al., 2011b as a robust alternative to the distance a compact set. In practice, the DTM can be estimated by its empirical counterpart, that is the distance to the empirical measure (DTEM). In this paper we give a tight control of the deviation of the DTEM. Our analysis relies on a local analysis of empirical processes. In particular, we show that the rate of convergence of the DTEM directly depends on the regularity at zero of a particular quantile function which contains some local information about the geometry of the support. This quantile function is the relevant quantity to describe precisely how difficult is a geometric inference problem. Several numerical experiments illustrate the convergence of the DTEM and also confirm that our bounds are tight.

Article information

Electron. J. Statist., Volume 10, Number 2 (2016), 2243-2286.

Received: March 2016
First available in Project Euclid: 25 August 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G30: Order statistics; empirical distribution functions 68U05: Computer graphics; computational geometry [See also 65D18] 62-07: Data analysis 28A33: Spaces of measures, convergence of measures [See also 46E27, 60Bxx]

Geometric inference distance to measure rates of convergence


Chazal, Frédéric; Massart, Pascal; Michel, Bertrand. Rates of convergence for robust geometric inference. Electron. J. Statist. 10 (2016), no. 2, 2243--2286. doi:10.1214/16-EJS1161. https://projecteuclid.org/euclid.ejs/1472125729

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  • Arias-Castro, E., Donoho, D., and Huo, X. (2006). Adaptive multiscale detection of filamentary structures in a background of uniform random points., The Annals of Statistics, 34:326–349.
  • Biau, G., Chazal, F., Cohen-Steiner, D., Devroye, L., and Rodriguez, C. (2011). A weighted k-nearest neighbor density estimate for geometric inference., Electronic Journal of Statistics, 5:204–237.
  • Bobkov, S. and Ledoux, M. (2014). One-dimensional empirical measures, order statistics and Kantorovich transport distances., Preprint.
  • Buchet, M., Chazal, F., Dey, T. K., Fan, F., Oudot, S. Y., and Wang, Y. (2015a). Topological analysis of scalar fields with outliers. In, Proc. Sympos. on Computational Geometry.
  • Buchet, M., Chazal, F., Oudot, S., and Sheehy, D. R. (2015b). Efficient and robust persistent homology for measures. In, Proceedings of the 26th ACM-SIAM symposium on Discrete algorithms. SIAM. SIAM.
  • Caillerie, C., Chazal, F., Dedecker, J., and Michel, B. (2011). Deconvolution for the Wasserstein metric and geometric inference., Electron. J. Stat., 5:1394–1423.
  • Cambanis, S., Simons, G., and Stout, W. (1976). Inequalities for $\mathbbEk(x,y)$ when the marginals are fixed., Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 36(4):285–294.
  • Carlsson, G. (2009). Topology and data., Bulletin of the American Mathematical Society, 46(2):255–308.
  • Chazal, F., Chen, D., Guibas, L., Jiang, X., and Sommer, C. (2011a). Data-driven trajectory smoothing. In, Proc. ACM SIGSPATIAL GIS.
  • Chazal, F., Cohen-Steiner, D., and Lieutier, A. (2009a). Normal cone approximation and offset shape isotopy., Computational Geometry, 42(6):566–581.
  • Chazal, F., Cohen-Steiner, D., Lieutier, A., and Thibert, B. (2009b). Stability of Curvature Measures., Computer Graphics Forum (proc. SGP 2009), pages 1485–1496.
  • Chazal, F., Cohen-Steiner, D., and Mérigot, Q. (2011b). Geometric inference for probability measures., Foundations of Computational Mathematics, 11(6):733–751.
  • Chazal, F., Fasy, B. T., Lecci, F., Michel, B., Rinaldo, A., and Wasserman, L. (2014a). Robust topological inference: Distance to a measure and kernel distance., arXiv preprint arXiv:1412.7197.
  • Chazal, F., Fasy, B. T., Lecci, F., Michel, B., Rinaldo, A., and Wasserman, L. (2014b). Subsampling methods for persistent homology., arXiv preprint 1406.1901, accepted for ICML15.
  • Chazal, F., Glisse, M., Labruère, C., and Michel, B. (2015). Convergence rates for persistence diagram estimation in topological data analysis., Journal of Machine Learning Research, 16:3603–3635.
  • Chazal, F., Guibas, L. J., Oudot, S. Y., and Skraba, P. (2013). Persistence-based clustering in riemannian manifolds., Journal of the ACM (JACM), 60(6):41.
  • Chazal, F. and Lieutier, A. (2008). Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees., Computational Geometry, 40(2):156–170.
  • Cuevas, A. (2009). Set estimation: another bridge between statistics and geometry., Bol. Estad. Investig. Oper., 25(2):71–85.
  • Cuevas, A. and Rodríguez-Casal, A. (2004). On boundary estimation., Advances in Applied Probability, pages 340–354.
  • del Barrio, E., Giné, E., and Matrán, C. (1999). The central limit theorem for the Wasserstein distance between the empirical and the true distributions., Ann. Probab., 27:1009–1971.
  • del Barrio, E., Giné, E., and Utzet, F. (2005). Asymptotics for $\mathbbL_2$ functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances., Bernoulli, 11:131–189.
  • Dereich, S., Scheutzow, M., and Schottstedt, R. (2013). Constructive quantization: Approximation by empirical measures., Ann. Inst. H. Poincaré Probab. Statist., 49:1183–1203.
  • Devroye, L. and Wise, G. L. (1980). Detection of abnormal behavior via nonparametric estimation of the support., SIAM Journal on Applied Mathematics, 38(3):480–488.
  • Dvoretzky, A., Kiefer, J., and Wolfowitz, J. (1956). Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator., The Annals of Mathematical Statistics, pages 642–669.
  • Fasy, B. T., Lecci, F., Rinaldo, A., Wasserman, L., Balakrishnan, S., Singh, A., et al. (2014). Confidence sets for persistence diagrams., The Annals of Statistics, 42(6):2301–2339.
  • Fournier, N. and Guillin, A. (2013). On the rate of convergence in wasserstein distance of the empirical measure., Probability Theory and Related Fields, pages 1–32.
  • Genovese, C., Perone-Pacifico, M., Verdinelli, I., and Wasserman, L. (2009). On the path density of a gradient field., The Annals of Statistics, 37:3236–3271.
  • Genovese, C. R., Perone-Pacifico, M., Verdinelli, I., and Wasserman, L. (2012). Manifold estimation and singular deconvolution under hausdorff loss., The Annals of Statistics, 40(2):941–963.
  • Guibas, L., Morozov, D., and Mérigot, Q. (2013). Witnessed k-distance., Discrete Comput. Geom., 49:22–45.
  • Hastie, T. and Stuetzle, W. (1989). Principal curves., J. Amer. Statist. Assoc., 84(406):502–516.
  • Mammen, E., Tsybakov, A. B., et al. (1999). Smooth discrimination analysis., The Annals of Statistics, 27(6):1808–1829.
  • Massart, P. (1990). The tight constant in the dvoretzky-kiefer-wolfowitz inequality., The Annals of Probability, 18(3):pp. 1269–1283.
  • Massart, P. (2007)., Concentration inequalities and model selection. Springer, Berlin. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003.
  • Niyogi, P., Smale, S., and Weinberger, S. (2008). Finding the homology of submanifolds with high confidence from random samples., Discrete & Computational Geometry, 39(1-3):419–441.
  • Phillips, J. M., Wang, B., and Zheng, Y. (2014). Geometric inference on kernel density estimates., arXiv preprint 1307.7760.
  • R Core Team (2014)., R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
  • Rachev, S. and Rüschendorf, L. (1998)., Mass transportation problems, volume II of Probability and its Applications. Springer-Verlag.
  • Shorack, G. R. and Wellner, J. A. (2009)., Empirical processes with applications to statistics, volume 59. SIAM.
  • Singh, A., Scott, C., and Nowak, R. (2009). Adaptive hausdorff estimation of density level sets., The Annals of Statistics, 37(5B):2760–2782.
  • Villani, C. (2008)., Optimal Transport: Old and New. Grundlehren Der Mathematischen Wissenschaften. Springer-Verlag.
  • Yu, B. (1997). Assouad, Fano, and Le Cam. In, Festschrift for Lucien Le Cam, pages 423–435. Springer, New York.