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 derive a non-asymptotic tail bound for the variable selection consistency of our method. The proposed method is illustrated through simulation studies and a real-data application from the Human Connectome Project.

翻译：在现代高频金融数据和神经影像数据分析等应用中，高维函数数据日益普遍。本文研究一类高维线性回归模型，其中每个预测变量均为无限维函数空间中的随机元素，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个再生核希尔伯特空间(RKHS)，我们通过在系数函数的RKHS范数上施加群弹性网型惩罚项来正则化模型拟合。研究表明，我们的损失函数是Gateaux次可微的，且函数型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的支持下，我们推导了该方法变量选择一致性的非渐近尾界。通过模拟研究与人类连接组项目实际数据应用，验证了所提方法的有效性。