# The Regularization Path for Smoothing Splines

I've got a potentially interesting question. Does anyone know if R already has a package for calculating the entire regularization path of the smoothing spline?

That is, for: $$\hat{f}_{\lambda}=argmin\|y-f\|^{2}+\lambda\int(f''(x))^2dx$$ Does R have a function that will calculate the path $\{\hat{f}_{\lambda},\forall \lambda \ge0 \}$ much like glmnet for the elastic net or svmpath for svms?

The problem is "solved" in the general sense in that Rosset, Zhu (2007) gave general conditions under which we can find that path. They even give the full algorithm for the locally adaptive regression splines (which is the same as above with a TV penalty instead) but nothing for the cubic spline.

I was told that GCV analytically finds the regularization path but as far as I can tell, GCV analytically finds the error as a function of the regularization parameter which, instead of giving us the full path, is useful for choosing the "best" (loaded word) value of $\lambda$.

Any word on whether there's a solver for this sort of problem?

• The gamsel package talks about fitting the "entire regularization path" both in the arxiv preprint article and the manual – Henry Oct 29 '16 at 23:21
• I actually ran into the gamsel package but I'm not entirely sure I can write the spline objective as the objective function they specify. It also specifies a particular basis. It does look interesting though. Have you been able to get it to compile? – Nick Thieme Oct 31 '16 at 17:58