  Smoothing splines provide a powerful and flexible means for nonparametric
estimation and inference. With a cubic time complexity, fitting smoothing
spline models to large data is computationally prohibitive. In this paper, we
use the theoretical optimal eigenspace to derive a low rank approximation of
the smoothing spline estimates. We develop a method to approximate the
eigensystem when it is unknown and derive error bounds for the approximate
estimates. The proposed methods are easy to implement with existing software.
Extensive simulations show that the new methods are accurate, fast, and
compares favorably against existing methods.
