$L_1$-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.

Classification : 62J02, 62G05, 62J07
Keywords: Functional regression, sparse recovery, LASSO, oracle inequality, infinite dictionaries
Mot clés : Régression fonctionnelle, recouvrement « sparse », LASSO, inégalité d’oracle, dictionnaire infini

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