Estimating composite functions by model selection

Yannick Baraud; Lucien Birgé

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 1, page 285-314
  • ISSN: 0246-0203

Abstract

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We consider the problem of estimating a function s on [ - 1 , 1 ] k for large values of k by looking for some best approximation of s by composite functions of the form g ∘ u . Our solution is based on model selection and leads to a very general approach to solve this problem with respect to many different types of functions g , u and statistical frameworks. In particular, we handle the problems of approximating s by additive functions, single and multiple index models, artificial neural networks, mixtures of Gaussian densities (when s is a density) among other examples. We also investigate the situation where s = g ∘ u for functions g and u belonging to possibly anisotropic smoothness classes. In this case, our approach leads to a completely adaptive estimator with respect to the regularities of g and u .

How to cite

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Baraud, Yannick, and Birgé, Lucien. "Estimating composite functions by model selection." Annales de l'I.H.P. Probabilités et statistiques 50.1 (2014): 285-314. <http://eudml.org/doc/272057>.

@article{Baraud2014,
abstract = {We consider the problem of estimating a function $s$ on $[-1,1]^\{k\}$ for large values of $k$ by looking for some best approximation of $s$ by composite functions of the form $g\circ u$. Our solution is based on model selection and leads to a very general approach to solve this problem with respect to many different types of functions $g,u$ and statistical frameworks. In particular, we handle the problems of approximating $s$ by additive functions, single and multiple index models, artificial neural networks, mixtures of Gaussian densities (when $s$ is a density) among other examples. We also investigate the situation where $s=g\circ u$ for functions $g$ and $u$ belonging to possibly anisotropic smoothness classes. In this case, our approach leads to a completely adaptive estimator with respect to the regularities of $g$ and $u$.},
author = {Baraud, Yannick, Birgé, Lucien},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {curve estimation; model selection; composite functions; adaptation; single index model; artificial neural networks; gaussian mixtures; Gaussian mixtures},
language = {eng},
number = {1},
pages = {285-314},
publisher = {Gauthier-Villars},
title = {Estimating composite functions by model selection},
url = {http://eudml.org/doc/272057},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Baraud, Yannick
AU - Birgé, Lucien
TI - Estimating composite functions by model selection
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 1
SP - 285
EP - 314
AB - We consider the problem of estimating a function $s$ on $[-1,1]^{k}$ for large values of $k$ by looking for some best approximation of $s$ by composite functions of the form $g\circ u$. Our solution is based on model selection and leads to a very general approach to solve this problem with respect to many different types of functions $g,u$ and statistical frameworks. In particular, we handle the problems of approximating $s$ by additive functions, single and multiple index models, artificial neural networks, mixtures of Gaussian densities (when $s$ is a density) among other examples. We also investigate the situation where $s=g\circ u$ for functions $g$ and $u$ belonging to possibly anisotropic smoothness classes. In this case, our approach leads to a completely adaptive estimator with respect to the regularities of $g$ and $u$.
LA - eng
KW - curve estimation; model selection; composite functions; adaptation; single index model; artificial neural networks; gaussian mixtures; Gaussian mixtures
UR - http://eudml.org/doc/272057
ER -

References

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  1. [1] N. Akakpo. Adaptation to anisotropy and inhomogeneity via dyadic piecewise polynomial selection. Math. Methods Statist.21 (2012) 1–28. Zbl1308.62070MR2901269
  2. [2] Y. Baraud. Estimator selection with respect to Hellinger-type risks. Probab. Theory Related Fields151 (2011) 353–401. Zbl05968717MR2834722
  3. [3] Y. Baraud, F. Comte and G. Viennet. Model selection for (auto-)regression with dependent data. ESAIM Probab. Stat.5 (2001) 33–49. Zbl0990.62035MR1845321
  4. [4] Y. Baraud, C. Giraud and S. Huet. Gaussian model selection with an unknown variance. Ann. Statist.37 (2009) 630–672. Zbl1162.62051MR2502646
  5. [5] A. R. Barron, L. Birgé and P. Massart. Risk bounds for model selection via penalization. Probab. Theory Related Fields113 (1999) 301–413. Zbl0946.62036MR1679028
  6. [6] A. R. Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inform. Theory39 (1993) 930–945. Zbl0818.68126MR1237720
  7. [7] A. R. Barron. Approximation and estimation bounds for artificial neural networks. Machine Learning14 (1994) 115–133. Zbl0818.68127
  8. [8] L. Birgé. Model selection via testing: An alternative to (penalized) maximum likelihood estimators. Ann. Inst. Henri Poincaré Probab. Stat.42 (2006) 273–325. Zbl1333.62094MR2219712
  9. [9] L. Birgé. Model selection for Poisson processes. In Asymptotics: Particles, Processes and Inverse Problems, Festschrift for Piet Groeneboom 32–64. E. Cator, G. Jongbloed, C. Kraaikamp, R. Lopuhaä and J. Wellner (Eds). IMS Lecture Notes – Monograph Series 55. Inst. Math. Statist., Beachwood, OH, 2007. Zbl1176.62082MR2459930
  10. [10] L. Birgé. Model selection for density estimation with 𝕃 2 -loss. Probab. Theory Related Fields. To appear. Available at http://arxiv.org/abs/1102.2818. Zbl1285.62037
  11. [11] L. Birgé and P. Massart. Gaussian model selection. J. Eur. Math. Soc. (JEMS) 3 (2001) 203–268. Zbl1037.62001MR1848946
  12. [12] W. Dahmen, R. DeVore and K. Scherer. Multidimensional spline approximation. SIAM J. Numer. Anal.17 (1980) 380–402. Zbl0437.41010MR581486
  13. [13] R. DeVore and G. Lorentz. Constructive Approximation. Springer, Berlin, 1993. Zbl0797.41016MR1261635
  14. [14] J. Friedman and J. Tukey. A projection pursuit algorithm for exploratory data analysis. IEEE Trans. Comput. C-23 (1974) 881–890. Zbl0284.68079
  15. [15] R. Hochmuth. Wavelet characterizations for anisotropic Besov spaces. Appl. Comput. Harmon. Anal.12 (2002) 179–208. Zbl1003.42024MR1884234
  16. [16] J. L. Horowitz and E. Mammen. Rate-optimal estimation for a general class of nonparametric regression models with unknown link functions. Ann. Statist.35 (2007) 2589–2619. Zbl1129.62034MR2382659
  17. [17] P. J. Huber. Projection pursuit (with discussion). Ann. Statist.13 (1985) 435–525. Zbl0595.62059MR790553
  18. [18] A. B. Juditsky, O. V. Lepski and A. B. Tsybakov. Nonparametric estimation of composite functions. Ann. Statist.37 (2009) 1360–1404. Zbl1160.62030MR2509077
  19. [19] C. Maugis and B. Michel. A non asymptotic penalized criterion for Gaussian mixture model selection. ESAIM Probab. Stat.15 (2011) 41–68. Zbl06157507MR2870505
  20. [20] C. J. Stone. Optimal global rates of convergence for nonparametric regression. Ann. Statist.10 (1982) 1040–1053. Zbl0511.62048MR673642

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