  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 of
predictors $d\geq 2$. Such a sizable 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 identical rate
when $q$ is of the order $O\{n^{2/(pr+1)}\}$, where $p\in [1,2]$ depends on the
true function $\eta$, and $r > 1$ depends on the type of spline. Such $q$ is
called the essential number of basis functions. In this article, we develop a
more efficient basis selection method. By selecting the ones corresponding to
roughly equal-spaced observations, the proposed method chooses a set of basis
functions with a large diversity. The asymptotic analysis shows our proposed
smoothing spline estimator can decrease $q$ to roughly $O\{n^{1/(pr+1)}\}$,
when $d\leq pr+1$. Applications on synthetic and real-world datasets show the
proposed method leads to a smaller prediction error compared with other basis
selection methods.
