# Lasso in infinite dimension: application to variable selection in functional multivariate linear regression

## TL;DR

This paper extends the Lasso method to infinite-dimensional functional data, proposing two estimation approaches for variable selection in multivariate functional linear regression, with theoretical guarantees and practical algorithms.

## Contribution

It introduces two Lasso-based estimation methods for functional data, providing oracle inequalities and a coordinate descent algorithm for implementation.

## Key findings

- The methods effectively select relevant covariates in simulated and real data.
- Oracle inequalities demonstrate theoretical sparsity guarantees.
- The coordinate descent algorithm efficiently computes solutions.

## Abstract

It is more and more frequently the case in applications that the data we observe come from one or more random variables taking values in an infinite dimensional space, e.g. curves. The need to have tools adapted to the nature of these data explains the growing interest in the field of functional data analysis. The model we study in this paper assumes a linear dependence between a quantity of interest and several covariates, at least one of which has an infinite dimension. To select the relevant covariates in this context, we investigate adaptations of the Lasso method. Two estimation methods are defined. The first one consists in the minimization of a Group-Lasso criterion on the multivariate functional space H. The second one minimizes the same criterion but on a finite dimensional subspaces of H whose dimension is chosen by a penalized least squares method. We prove oracle inequalities of sparsity in the case where the design is fixed or random. To compute the solutions of both criteria in practice, we propose a coordinate descent algorithm. A numerical study on simulated and real data illustrates the behavior of the estimators.