High-dimensional Hilbert–Schmidt linear regression with Hilbert manifold variables
提出了一种处理希尔伯特流形上变量间线性回归的方法，适用于响应和协变量来自不同空间且协变量数量随样本量指数增长的情形，通过非凸惩罚实现变量选择，并给出了理论性质和计算算法。
In this paper, we propose a novel high-dimensional linear regression technique for variables taking values in Hilbert manifolds. Our approach offers a flexible framework where the response and covariates originate from distinct spaces and are interconnected by Hilbert–Schmidt operators. Our methodology is designed for situations where the number of the covariates grows exponentially fast as the sample size increases and some of the covariates together with the response take values in infinite dimensional spaces. It is formulated under a general penalization scheme that includes various nonconvex penalty functions. Leveraging modern statistical theory for data residing on Hilbert manifolds, we establish the oracle property and derive error bounds for the proposed estimators. We also provide an efficient computational algorithm to solve the associated constrained optimization problem. The practical performance of the proposed method is demonstrated via numerical simulation and real data applications.