High-dimensional functional data has become increasingly prevalent in modern applications such as high-frequency financial data and neuroimaging data analysis. We investigate a class of high-dimensional linear regression models, where each predictor is a random element in an infinite dimensional function space, and the number of functional predictors p can potentially be much greater than the sample size n. Assuming that each of the unknown coefficient functions belongs to some reproducing kernel Hilbert space (RKHS), we regularized the fitting of the model by imposing a group elastic-net type of penalty on the RKHS norms of the coefficient functions. We show that our loss function is Gateaux sub-differentiable, and our functional elastic-net estimator exists uniquely in the product RKHS. Under suitable sparsity assumptions and a functional version of the irrepresentible condition, we establish the variable selection consistency property of our approach. The proposed method is illustrated through simulation studies and a real-data application from the Human Connectome Project.

翻译：高维函数数据在高频金融数据与神经影像数据分析等现代应用中日益常见。本文研究一类高维线性回归模型，其中每个预测变量为无穷维函数空间中的随机元，且函数型预测变量个数p可能远大于样本量n。假设每个未知系数函数属于某再生核希尔伯特空间(RKHS)，我们通过对系数函数的RKHS范数施加群弹性网型惩罚来正则化模型拟合。研究表明损失函数具有Gateaux次可微性，且函数弹性网估计量在乘积RKHS中唯一存在。在适当稀疏性假设及函数型不可表示条件成立下，我们建立了该方法变量选择的一致性性质。通过模拟实验及人类连接组项目的实际数据应用验证了所提方法的有效性。