More efficient approximation of smoothing splines via space-filling basis selection

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 |$d\leq pr+1$|⁠. Applications to synthetic and real-world datasets show that the proposed method leads to a smaller prediction error than other basis selection methods.