$L_1$-penalization in functional linear regression with subgaussian design

Koltchinskii, Vladimir; Minsker, Stanislav

Journal de l’École polytechnique - Mathématiques, Tome 1 (2014), pp. 269-330.

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.

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

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

[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. ${\ell }_1$-regularized linear regression: persistence and oracle inequalities, Probab. Theory Relat. Fields, Volume 154 (2012), pp. 193-224 | MR

[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., Volume 37 (2009) no. 4, pp. 1705-1732 | MR | Zbl

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

[8] Bühlmann, P.; van de Geer, S. A. Statistics for high-dimensional data, Springer- Verlag, Berlin-Heidelberg, 2011 | MR | Zbl

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

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

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

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

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

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

[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., Volume 36 (2008) no. 2, pp. 614-645 | MR | Zbl

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

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

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

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

[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

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

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

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

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

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

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

[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, Graduate Texts in Math., 142, Springer, 1993 | MR | Zbl

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

[31] Massart, P.; Meynet, C. The Lasso as an ${\ell }_1$-ball model selection procedure, Electron. J. Statist., Volume 5 (2011), pp. 669-687 | MR | Zbl

[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 ${\psi }_1$ diameter, Geom. Funct. Anal., Volume 20 (2010) no. 4, pp. 988-1027 | MR | Zbl

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

[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 Series in Statistics, 77, Springer, New York, 2002 | MR | Zbl

[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 | Zbl

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

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

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

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

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