In this paper, we establish minimax optimal rates of convergence for prediction in a semi-functional linear model that consists of a functional component and a less smooth nonparametric component. Our results reveal that the smoother functional component can be learned with the minimax rate as if the nonparametric component were known. More specifically, a double-penalized least squares method is adopted to estimate both the functional and nonparametric components within the framework of reproducing kernel Hilbert spaces. By virtue of the representer theorem, an efficient algorithm that requires no iterations is proposed to solve the corresponding optimization problem, where the regularization parameters are selected by the generalized cross validation criterion. Numerical studies are provided to demonstrate the effectiveness of the method and to verify the theoretical analysis.

翻译：在本文中,我们在半功能线性模型中为预测设定了最低最佳合差率,该模型由功能部分和不那么平滑的非参数部分组成。我们的结果显示,可以通过微型通融率学习更顺畅的功能部分,仿佛了解非参数部分。更具体地说,采用了一种双重处罚最低平方法,在复制核心部分Hilbert空间的框架内估算功能和非参数部分。根据代表理论,建议一种不需要迭代的有效算法,以解决相应的优化问题,因为正规化参数是通过通用交叉验证标准选择的。提供了数字研究,以证明该方法的有效性并核实理论分析。