$L_1$-Penalization in Functional Linear Regression with Subgaussian Design

TL;DR

This paper investigates $L_1$-penalized functional linear regression with subgaussian design, focusing on sparse slope functions represented as a few well-separated spikes, and establishes sharp oracle inequalities for the estimator.

Contribution

It extends sparse estimation to infinite dictionaries in functional regression and introduces parameters characterizing sparsity with theoretical error bounds.

Findings

Derived sharp oracle inequalities for the $L_2$-error.

Demonstrated the estimator's effectiveness in sparse functional regression.

Provided theoretical insights into sparsity parameters and estimator performance.

Abstract

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.