Ex. 5.8

In the appendix to this chapter we show how the smoothing spline computations could be more efficiently carried out using a \((N + 4)\) dimensional basis of B-splines. Describe a slightly simpler scheme using a \((N +2)\) dimensional B-spline basis defined on the \(N-2\) interior knots.

Soln. 5.8

I believe the relevant text is in the last section of Appendix in Chapter 5. The use of B-splines reduces the complexity from \(O(N^3)\) to \(O(N)\) via Cholesky decomposition of a 4-banded matrix \((\textbf{B}^T\textbf{B}+\lambda \bm{\Omega})\). It seems to me, using \((N+2)\) dimensional B-splines yields \(\textbf{B}\in \mathbb{R}^{N\times (N+2)}\) and \(\bm{\Omega}\in\mathbb{R}^{(N+2)\times (N+2)}\), which is slightly simpler than \(N\times (N+4)\) and \((N+4)\times (N+4)\) matrices, however it's unclear to me what the essential differences are.