# Adaptive Function-on-Scalar Regression with a Smoothing Elastic Net

## TL;DR

This paper introduces AFSSEN, a novel adaptive Elastic Net approach for high-dimensional function-on-scalar regression that effectively selects predictors and ensures smooth estimates within a Hilbert space framework.

## Contribution

The paper develops a new regularization method combining functional norms for simultaneous variable selection and smoothing in a high-dimensional setting.

## Key findings

- AFSSEN outperforms existing methods in prediction accuracy.
- It achieves better variable selection with fewer false positives.
- Theoretical properties include a functional oracle property.

## Abstract

This paper presents a new methodology, called AFSSEN, to simultaneously select significant predictors and produce smooth estimates in a high-dimensional function-on-scalar linear model with a sub-Gaussian errors. Outcomes are assumed to lie in a general real separable Hilbert space, H, while parameters lie in a subspace known as a Cameron Martin space, K, which are closely related to Reproducing Kernel Hilbert Spaces, so that parameter estimates inherit particular properties, such as smoothness or periodicity, without enforcing such properties on the data. We propose a regularization method in the style of an adaptive Elastic Net penalty that involves mixing two types of functional norms, providing a fine tune control of both the smoothing and variable selection in the estimated model. Asymptotic theory is provided in the form of a functional oracle property, and the paper concludes with a simulation study demonstrating the advantage of using AFSSEN over existing methods in terms of prediction error and variable selection.