This paper considers the partially functional linear model (PFLM) where all predictive features consist of a functional covariate and a high dimensional scalar vector. Over an infinite dimensional reproducing kernel Hilbert space, the proposed estimation for PFLM is a least square approach with two mixed regularizations of a function-norm and an $\ell_1$-norm. Our main task in this paper is to establish the minimax rates for PFLM under high dimensional setting, and the optimal minimax rates of estimation is established by using various techniques in empirical process theory for analyzing kernel classes. In addition, we propose an efficient numerical algorithm based on randomized sketches of the kernel matrix. Several numerical experiments are implemented to support our method and optimization strategy.
翻译:本文件考虑了部分功能线性模型(PFLM),其中所有预测特征都包含功能共变和高维卡路里矢量。在一个无限的维度再生产内核Hilbert空间,对PFLM的拟议估算是一种最小的平方法,有两种功能-诺姆和1美元-诺姆的混合规范。我们本文件的主要任务是在高维设置下为PFLM确定迷你速率,而最优的小型估算速率是通过使用分析内核等级的经验过程理论的各种技术来确定的。此外,我们根据随机绘制的内核矩阵草图,提出了高效的数字算法。为了支持我们的方法和优化战略,我们进行了数项实验。