More efficient approximation of smoothing splines via space-filling basis selection. (7th May 2020) - Meng, Cheng; Zhang, Xinlian; Zhang, Jingyi; Zhong, Wenxuan; Ma, Ping
Summary: We consider the problem of approximating smoothing spline estimators in a nonparametric regression model. When applied to a sample of size $n$, the smoothing spline estimator can be expressed as a linear combination of $n$ basis functions, requiring $O(n^3)$ computational time when the number $d$ of predictors is two or more. Such a sizeable computational cost hinders the broad applicability of smoothing splines. In practice, the full-sample smoothing spline estimator can be approximated by an estimator based on $q$ randomly selected basis functions, resulting in a computational cost of $O(nq^2)$ . It is known that these two estimators converge at the same rate when $q$ is of order $O\{n^{2/(pr+1)}\}$, where $p\in [1, 2]$ depends on the true function and $r > 1$ depends on the type of spline. Such a $q$ is called the essential number of basis functions. In this article, we develop a more efficient basis selection method. By selecting basis functions corresponding to approximately equally spaced observations, the proposed method chooses a set of basis functions with great diversity. The asymptotic analysis shows that the proposed smoothing spline estimator can decrease $q$ to around $O\{n^{1/(pr+1)}\}$ when $d≤ pr+1$ . Applications to synthetic and real-world datasets show that the proposed method leads to a smaller prediction error than other basis selection methods.