Variable Selection and Minimax Prediction in High-dimensional Functional Linear Model

• Main Authors: Guo, Xingche, Li, Yehua, Hsing, Tailen
• Format: Journal Article
• Language: English
• Published: 22.10.2023
• Subjects: Mathematics - Statistics Theory; Statistics - Methodology; Statistics - Theory
• DOI: 10.48550/arxiv.2310.14419

High-dimensional functional data have 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 ultra-high. Assuming that each of the unknown coefficient functions belongs to some reproducing kernel Hilbert space (RKHS), we regularize 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 irrepresentable condition, we derive a non-asymptotic tail bound for variable selection consistency of our method. Allowing the number of true functional predictors $q$ to diverge with the sample size, we also show a post-selection refined estimator can achieve the oracle minimax optimal prediction rate. The proposed methods are illustrated through simulation studies and a real-data application from the Human Connectome Project.