Kernel and wavelet density estimators on manifolds and more general metric spaces. (English) Zbl 1439.62097

Summary: We consider the problem of estimating the density of observations taking values in classical or nonclassical spaces such as manifolds and more general metric spaces. Our setting is quite general but also sufficiently rich in allowing the development of smooth functional calculus with well localized spectral kernels, Besov regularity spaces, and wavelet type systems. Kernel and both linear and nonlinear wavelet density estimators are introduced and studied. Convergence rates for these estimators are established and discussed.


62G07 Density estimation
62R30 Statistics on manifolds
62R20 Statistics on metric spaces
35K08 Heat kernel
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text: DOI arXiv Euclid


[1] Baldi, P., Kerkyacharian, G., Marinucci, D. and Picard, D. (2009). Adaptive density estimation for directional data using needlets. Ann. Statist. 37 3362-3395. · Zbl 1369.62061
[2] Bhattacharya, A. and Bhattacharya, R. (2012). Nonparametric Inference on Manifolds: With Applications to Shape Spaces. Institute of Mathematical Statistics (IMS) Monographs 2. Cambridge: Cambridge Univ. Press. · Zbl 1273.62014
[3] Castillo, I., Kerkyacharian, G. and Picard, D. (2014). Thomas Bayes’ walk on manifolds. Probab. Theory Related Fields 158 665-710. · Zbl 1285.62028
[4] Cleanthous, G., Georgiadis, A.G., Kerkyacharian, G., Petrushev, P. and Picard, D. (2020). Supplement to “Kernel and wavelet density estimators on manifolds and more general metric spaces.” https://doi.org/10.3150/19-BEJ1171SUPP.
[5] Coifman, R.R. and Weiss, G. (1971). Analyse Harmonique Non-commutative sur Certains Espaces Homogènes: Étude de certaines intégrales singulières. Lecture Notes in Mathematics 242. Berlin: Springer. · Zbl 0224.43006
[6] Coulhon, T., Kerkyacharian, G. and Petrushev, P. (2012). Heat kernel generated frames in the setting of Dirichlet spaces. J. Fourier Anal. Appl. 18 995-1066. · Zbl 1270.58015
[7] Dai, F. and Xu, Y. (2013). Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics. New York: Springer. · Zbl 1275.42001
[8] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539. · Zbl 0860.62032
[9] Faraut, J. (2008). Analysis on Lie Groups: An Introduction. Cambridge Studies in Advanced Mathematics 110. Cambridge: Cambridge Univ. Press.
[10] Folland, G.B. (1999). Real Analysis: Modern Techniques and Their Applications, 2nd ed. Pure and Applied Mathematics (New York). New York: Wiley. A Wiley-Interscience Publication. · Zbl 0924.28001
[11] Frazier, M. and Jawerth, B. (1985). Decomposition of Besov spaces. Indiana Univ. Math. J. 34 777-799. · Zbl 0551.46018
[12] Frazier, M. and Jawerth, B. (1990). A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93 34-170. · Zbl 0716.46031
[13] Frazier, M., Jawerth, B. and Weiss, G. (1991). Littlewood-Paley Theory and the Study of Function Spaces. CBMS Regional Conference Series in Mathematics 79. Providence, RI: Amer. Math. Soc. · Zbl 0757.42006
[14] Georgiadis, A.G., Kerkyacharian, G., Kyriazis, G. and Petrushev, P. (2017). Homogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators. J. Math. Anal. Appl. 449 1382-1412. · Zbl 1362.30052
[15] Goldenshluger, A. and Lepski, O. (2014). On adaptive minimax density estimation on \(R^d\). Probab. Theory Related Fields 159 479-543. · Zbl 1342.62053
[16] Grigor’yan, A. (2009). Heat Kernel and Analysis on Manifolds. AMS/IP Studies in Advanced Mathematics 47. Providence, RI: Amer. Math. Soc.
[17] Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation, and Statistical Applications. Lecture Notes in Statistics 129. New York: Springer. · Zbl 0899.62002
[18] Johnstone, I.M. and Silverman, B.W. (1990). Speed of estimation in positron emission tomography and related inverse problems. Ann. Statist. 18 251-280. · Zbl 0699.62043
[19] Juditsky, A. and Lambert-Lacroix, S. (2004). On minimax density estimation on \(\Bbb{R} \). Bernoulli 10 187-220. · Zbl 1076.62037
[20] Kerkyacharian, G. and Petrushev, P. (2015). Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces. Trans. Amer. Math. Soc. 367 121-189. · Zbl 1321.58017
[21] Kerkyacharian, G., Petrushev, P., Picard, D. and Willer, T. (2007). Needlet algorithms for estimation in inverse problems. Electron. J. Stat. 1 30-76. · Zbl 1320.62072
[22] Kerkyacharian, G., Petrushev, P. and Xu, Y. (2020). Gaussian bounds for the weighted heat kernels on the interval, ball and simplex. Constr. Approx. 51 73-122. · Zbl 1447.35166
[23] Kerkyacharian, G., Petrushev, P. and Xu, Y. (2020). Gaussian bounds for the heat kernels on the ball and the simplex: Classical approach. Studia Math. 250 235-252. · Zbl 1432.35108
[24] Kerkyacharian, G., Pham Ngoc, T.M. and Picard, D. (2011). Localized spherical deconvolution. Ann. Statist. 39 1042-1068. · Zbl 1216.62059
[25] Kerkyacharian, G. and Picard, D. (1992). Density estimation in Besov spaces. Statist. Probab. Lett. 13 15-24. · Zbl 0749.62026
[26] Lepski, O.V., Mammen, E. and Spokoiny, V.G. (1997). Optimal spatial adaptation to inhomogeneous smoothness: An approach based on kernel estimates with variable bandwidth selectors. Ann. Statist. 25 929-947. · Zbl 0885.62044
[27] Lepskiĭ, O.V. (1991). Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates. Teor. Veroyatn. Primen. 36 645-659. · Zbl 0738.62045
[28] Pelletier, B. (2005). Kernel density estimation on Riemannian manifolds. Statist. Probab. Lett. 73 297-304. · Zbl 1065.62063
[29] Pelletier, B. (2006). Non-parametric regression estimation on closed Riemannian manifolds. J. Nonparametr. Stat. 18 57-67. · Zbl 1088.62053
[30] Pollard, D. (1984). Convergence of Stochastic Processes. Springer Series in Statistics. New York: Springer. · Zbl 0544.60045
[31] Sandryhaila, A. and Moura, J.M.F. (2013). Discrete signal processing on graphs. IEEE Trans. Signal Process. 61 1644-1656. · Zbl 1393.94424
[32] Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis. Monographs on Statistics and Applied Probability. London: CRC Press. · Zbl 0617.62042
[33] Starck, J.-L., Murtagh, F. and Fadili, J.M. (2010). Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity. Cambridge: Cambridge Univ. Press. · Zbl 1196.94008
[34] Stein, E.M. and Weiss, G. (1971). Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series 32. Princeton, NJ: Princeton Univ. Press. · Zbl 0232.42007
[35] Tsybakov, A.B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. New York: Springer. Revised and extended from the 2004 French original, translated by Vladimir Zaiats.
[36] von Luxburg, U. (2007). A tutorial on spectral clustering. Stat. Comput. 17 395-416.
[37] Yosida, K. (1980). Functional Analysis, 6th ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 123. Berlin: Springer. · Zbl 0435.46002
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