L 1 -penalization in functional linear regression with subgaussian design
[Pénalisation L 1 en régression fonctionnelle linéaire avec design sous-gaussien]
Koltchinskii, Vladimir ; Minsker, Stanislav
Journal de l'École polytechnique - Mathématiques, Tome 1 (2014), p. 269-330 / Harvested from Numdam
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Nous étudions la régression fonctionnelle linéaire avec design sous-gaussien et la réponse à valeurs réelles. Nous nous concentrons sur les problèmes où la fonction de régression est bien approchée par un modèle fonctionnel linéaire dont la pente est « sparse » dans le sens où elle peut être représentée comme une somme d’un petit nombre de « pics » séparés. Nous pouvons considérer ce problème comme une extension du problème classique d’estimation « sparse » au cas d’un dictionnaire infini. Nous étudions un estimateur de la fonction de régression basé sur la minimisation du risque empirique pénalisé avec une perte quadratique et avec une pénalité de complexité définie en termes de la norme L 1 (une version continue du LASSO). L’objectif principal est d’introduire certains paramètres importants qui caractérisent la « sparsité » dans cette classe de problèmes et de prouver des inégalités d’oracle « sparses » montrant comment l’erreur L 2 de la version continue du LASSO dépend de la sparsité sous-jacent du problème.

We study functional regression with random subgaussian design and real-valued response. The focus is on the problems in which the regression function can be well approximated by a functional linear model with the slope function being “sparse” in the sense that it can be represented as a sum of a small number of well separated “spikes”. This can be viewed as an extension of now classical sparse estimation problems to the case of infinite dictionaries. We study an estimator of the regression function based on penalized empirical risk minimization with quadratic loss and the complexity penalty defined in terms of L 1 -norm (a continuous version of LASSO). The main goal is to introduce several important parameters characterizing sparsity in this class of problems and to prove sharp oracle inequalities showing how the L 2 -error of the continuous LASSO estimator depends on the underlying sparsity of the problem.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/jep.11
Classification:  62J02,  62G05,  62J07
Mots clés: Régression fonctionnelle, recouvrement « sparse », LASSO, inégalité d’oracle, dictionnaire infini 
@article{JEP_2014__1__269_0,
     author = {Koltchinskii, Vladimir and Minsker, Stanislav},
     title = {$L\_1$-penalization in functional linear regression with subgaussian design},
     journal = {Journal de l'\'Ecole polytechnique - Math\'ematiques},
     volume = {1},
     year = {2014},
     pages = {269-330},
     doi = {10.5802/jep.11},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEP_2014__1__269_0}
}

Koltchinskii, Vladimir; Minsker, Stanislav. $L_1$-penalization in functional linear regression with subgaussian design. Journal de l'École polytechnique - Mathématiques, Tome 1 (2014) pp. 269-330. doi : 10.5802/jep.11. http://gdmltest.u-ga.fr/item/JEP_2014__1__269_0/

[1] Adamczak, R. A tail inequality for suprema of unbounded empirical processes with applications to Markov chains, Electron. J. Probab., Tome 13 (2008), pp. 1000-1034 | MR 2424985 | Zbl 1190.60010

[2] Adams, R. Sobolev spaces, Academic Press, New York (1975) | MR 450957 | Zbl 1098.46001

[3] Bal, G. Numerical methods for PDEs (2009) (Lecture notes available at http://www.columbia.edu/~gb2030/COURSES/E6302/NumAnal.pdf)

[4] Bartlett, P. L.; Mendelson, S.; Neeman, J. ℓ 1 -regularized linear regression: persistence and oracle inequalities, Probab. Theory Relat. Fields, Tome 154 (2012), pp. 193-224 | MR 2981422

[5] Bednorz, W. Concentration via chaining method and its applications (2014) (arXiv:1405.0676v2)

[6] Bickel, P. J.; Ritov, Y.; Tsybakov, A. B. Simultaneous analysis of Lasso and Dantzig selector, Ann. Statist., Tome 37 (2009) no. 4, pp. 1705-1732 | MR 2533469 | Zbl 1173.62022

[7] Bogachev, V. I. Measure theory. Vol. I, II, Springer-Verlag, Berlin (2007), pp. Vol. I: xviii+500 pp., Vol. II: xiv+575 | MR 2267655 | Zbl 1120.28001

[8] Bühlmann, P.; Van De Geer, S. A. Statistics for high-dimensional data, Springer- Verlag, Berlin-Heidelberg (2011) | MR 2807761 | Zbl 1273.62015

[9] Bunea, F.; Tsybakov, A. B.; Wegkamp, M. Sparsity oracle inequalities for the Lasso, Electron. J. Statist., Tome 1 (2007), pp. 169-194 | MR 2312149 | Zbl 1146.62028

[10] Cai, T. T.; Hall, P. Prediction in functional linear regression, Ann. Statist., Tome 34 (2006) no. 5, pp. 2159-2179 | MR 2291496 | Zbl 1106.62036

[11] Candès, E. The restricted isometry property and its implications for compressed sensing, Comptes Rendus Mathématique, Tome 346 (2008) no. 9, pp. 589-592 | MR 2412803 | Zbl 1153.94002

[12] Candès, E.; Fernandez-Granda, C. Towards a Mathematical Theory of Super-resolution, Comm. Pure Appl. Math., Tome 67 (2014) no. 6, pp. 906-956 | MR 3193963

[13] Candès, E. J.; Romberg, J. K.; Tao, T. Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math., Tome 59 (2006) no. 8, pp. 1207-1223 | MR 2230846 | Zbl 1098.94009

[14] Crambes, C.; Kneip, A.; Sarda, P. Smoothing splines estimators for functional linear regression, Ann. Statist., Tome 37 (2009) no. 1, pp. 35-72 | MR 2488344 | Zbl 1169.62027

[15] Dirksen, S. Tail bounds via generic chaining (2013) (arXiv:1309.3522)

[16] Van De Geer, S. A. High-dimensional generalized linear models and the Lasso, Ann. Statist., Tome 36 (2008) no. 2, pp. 614-645 | MR 2396809 | Zbl 1138.62323

[17] Van De Geer, S. A.; Lederer, J. The Lasso, correlated design, and improved oracle inequalities, A Festschrift in Honor of Jon Wellner, Institute of Mathematical Statistics (IMS Collections) (2012), pp. 3468-3497 | MR 3202642

[18] Gluskin, E. D. Norms of random matrices and widths of finite-dimensional sets, Mat. Sb., Tome 120(162) (1983) no. 2, pp. 180-189 | MR 687610 | Zbl 0528.46015

[19] Hebiri, M.; Lederer, J. How Correlations Influence Lasso Prediction, IEEE Trans. Information Theory, Tome 59 (2013) no. 3, pp. 1846-1854 | Article | MR 3030757

[20] Ioffe, A. D.; Tikhomirov, V. M. Theory of Extremal Problems, Nauka, Moscow (1974) | MR 410502

[21] James, G. Sparseness and functional data analysis, The Oxford handbook of functional data analysis, Oxford University Press, New York (2011), pp. 298-323 | MR 2908027

[22] James, G. M.; Wang, J.; Zhu, J. Functional linear regression that’s interpretable, Ann. Statist., Tome 37 (2009) no. 5A, pp. 2083-2108 | MR 2543686 | Zbl 1171.62041

[23] Koltchinskii, V. The Dantzig selector and sparsity oracle inequalities, Bernoulli, Tome 15 (2009) no. 3, pp. 799-828 | MR 2555200

[24] Koltchinskii, V. Sparse recovery in Convex Hulls via Entropy penalization, Ann. Statist., Tome 37 (2009) no. 3, pp. 1332-1359 | MR 2509076 | Zbl 1269.62039

[25] Koltchinskii, V. Sparsity in Penalized Empirical Risk Minimization, Ann. Inst. H. Poincaré Probab. Statist., Tome 45 (2009) no. 1, pp. 7-57 | Numdam | MR 2500227 | Zbl 1168.62044

[26] Koltchinskii, V. Oracle inequalities in empirical risk minimization and sparse recovery problems, 38th Probability Summer School (Saint-Flour, 2008), Springer (2011) | MR 2829871 | Zbl 1223.91002

[27] Koltchinskii, V.; Lounici, K.; Tsybakov, A. B. Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion, Ann. Statist., Tome 39 (2011) no. 5, pp. 2302-2329 | MR 2906869 | Zbl 1231.62097

[28] Koltchinskii, V.; Minsker, S. Sparse Recovery in Convex Hulls of Infinite Dictionaries, COLT 2010, 23rd Conference on Learning Theory (2010), pp. 420-432

[29] Lang, S. Real and functional analysis, Springer, Graduate Texts in Math., Tome 142 (1993) | MR 1216137 | Zbl 0831.46001

[30] Lifshits, M. A. Gaussian random functions, Kluwer Academic Publishers, Dordrecht, Mathematics and its Applications, Tome 322 (1995) | MR 1472736 | Zbl 0832.60002

[31] Massart, P.; Meynet, C. The Lasso as an ℓ 1 -ball model selection procedure, Electron. J. Statist., Tome 5 (2011), pp. 669-687 | MR 2820635 | Zbl 1274.62468

[32] Mendelson, S. Oracle inequalities and the isomorphic method (Preprint, 2012. Available at http://maths-people.anu.edu.au/~mendelso/papers/subgaussian-12-01-2012.pdf)

[33] Mendelson, S. Empirical processes with a bounded ψ 1 diameter, Geom. Funct. Anal., Tome 20 (2010) no. 4, pp. 988-1027 | MR 2729283 | Zbl 1204.60042

[34] Müller, H. G.; Stadtmüller, U. Generalized functional linear models, Ann. Statist., Tome 33 (2005) no. 2, pp. 774-805 | Zbl 1068.62048

[35] Ramsay, J. O. Functional data analysis, Wiley Online Library (2006)

[36] Ramsay, J. O.; Silverman, B. W. Applied functional data analysis: methods and case studies, Springer, New York, Springer Series in Statistics, Tome 77 (2002) | MR 1910407 | Zbl 0882.62002

[37] Ritter, K.; Wasilkowski, G. W.; Woźniakowski, H. Multivariate integration and approximation for random fields satisfying Sacks-Ylvisaker conditions, Ann. Appl. Probab. (1995), pp. 518-540 | MR 1336881 | Zbl 0872.62063

[38] Sacks, J.; Ylvisaker, D. Designs for regression problems with correlated errors, Ann. Statist., Tome 37 (1966) no. 1, pp. 66-89 | MR 192601 | Zbl 0152.17503

[39] Talagrand, M. The generic chaining, Springer-Verlag, Berlin, Springer Monographs in Mathematics (2005) | MR 2133757 | Zbl 1075.60001

[40] Tibshirani, R. Regression shrinkage and selection via the Lasso, J. R. Stat. Soc. Ser. B Stat. Methodol. (1996), pp. 267-288 | MR 1379242 | Zbl 0850.62538

[41] Van Der Vaart, A. W.; Wellner, J. A. Weak convergence and empirical processes, Springer-Verlag, New York, Springer Series in Statistics (1996), pp. xvi+508 | MR 1385671 | Zbl 0862.60002

[42] Yuan, M.; Cai, T. T. A reproducing kernel Hilbert space approach to functional linear regression, Ann. Statist., Tome 38 (2010) no. 6, pp. 3412-3444 | MR 2766857 | Zbl 1204.62074