Efficient basis selection for smoothing splines via rotated lattices

Fitting a smoothing spline model on a large-scale dataset is daunting due to the high computational cost. In this study, we develop an efficient basis selection method for smoothing spline calculation. The key idea is to force a nonparametric function in an infinite-dimensional functional space to reside in a relatively small and finite-dimensional model space without the loss of too much prediction accuracy. Such an approximation naturally allows for much faster numerical calculation, especially for large datasets. Among various basis selection methods, space-filling basis selection has been proven to be more efficient since its model space dimension is smaller than that of others. Despite algorithmic benefits, most of the space-filling selection methods only take the overall space-filling property into account. These methods may be less efficient when the underlying response surface is not isomorphic. To overcome this obstacle, we develop an efficient algorithm to improve projective uniformity for space-filling basis selection. It has been proved that the proposed estimator has the same convergence rate as the full bases estimator. Compared with the standard approach, the proposed method significantly reduces the computational cost. Simulation and real data studies demonstrate the efficiency and superiority of the proposed method.