Rates of convergence for regression with the graph poly-Laplacian.

In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset {xi}i=1n and a set of noisy labels {yi}i=1n⊂R we let un:{xi}i=1n→R be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When yi=g(xi)+ξi, for iid noise ξi, and using the geometric random graph, we identify (with high probability) the rate of convergence of un to g in the large data limit n→∞. Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model.