Functional regression analysis is an established tool for many contemporary scientific applications. Regression problems involving large and complex data sets are ubiquitous, and feature selection is crucial for avoiding overfitting and achieving accurate predictions. We propose a new, flexible, and ultra-efficient approach to perform feature selection in a sparse high dimensional function-on-function regression problem, and we show how to extend it to the scalar-on-function framework. Our method combines functional data, optimization, and machine learning techniques to perform feature selection and parameter estimation simultaneously. We exploit the properties of Functional Principal Components, and the sparsity inherent to the Dual Augmented Lagrangian problem to significantly reduce computational cost, and we introduce an adaptive scheme to improve selection accuracy. Through an extensive simulation study, we benchmark our approach to the best existing competitors and demonstrate a massive gain in terms of CPU time and selection performance without sacrificing the quality of the coefficients' estimation. Finally, we present an application to brain fMRI data from the AOMIC PIOP1 study.

翻译：函数回归分析是当代众多科学应用中的既定工具。涉及大规模复杂数据集的回归问题普遍存在，特征选择对于避免过拟合和实现精确预测至关重要。我们提出了一种新型、灵活且超高效的方法，用于解决稀疏高维函数对函数回归问题中的特征选择，并展示了如何将其扩展至标量对函数框架。该方法融合了函数数据分析、优化和机器学习技术，同步实现特征选择与参数估计。我们利用函数主成分的特性以及对偶增广拉格朗日问题固有的稀疏性，显著降低了计算成本，并引入自适应机制以提高选择精度。通过广泛的模拟研究，我们将该方法与现有最优竞争方法进行基准测试，证明在不牺牲系数估计质量的前提下，在CPU时间和选择性能上实现了巨大提升。最后，我们将其应用于AOMIC PIOP1研究的脑功能磁共振成像数据。