Non-asymptotic Adaptive Prediction in Functional Linear Models

TL;DR

This paper introduces a non-asymptotic, adaptive estimation method for functional linear regression that automatically selects the model dimension using penalized least squares, achieving optimal convergence rates.

Contribution

It develops a novel adaptive estimation procedure based on penalized model selection for functional linear models, handling data-dependent eigenbasis for improved prediction accuracy.

Findings

Estimator satisfies an oracle inequality for prediction risk.

Achieves minimax optimal rates over ellipsoids.

Numerical comparison shows competitive performance with cross-validation.

Abstract

Functional linear regression has recently attracted considerable interest. Many works focus on asymptotic inference. In this paper we consider in a non asymptotic framework a simple estimation procedure based on functional Principal Regression. It revolves in the minimization of a least square contrast coupled with a classical projection on the space spanned by the m first empirical eigenvectors of the covariance operator of the functional sample. The novelty of our approach is to select automatically the crucial dimension m by minimization of a penalized least square contrast. Our method is based on model selection tools. Yet, since this kind of methods consists usually in projecting onto known non-random spaces, we need to adapt it to empirical eigenbasis made of data-dependent - hence random - vectors. The resulting estimator is fully adaptive and is shown to verify an oracle inequality for the risk associated to the prediction error and to attain optimal minimax rates of convergence over a certain class of ellipsoids. Our strategy of model selection is finally compared numerically with cross-validation.