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 $$ and a set of noisy labels $$ we let $$ be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When $$, for iid noise $$, and using the geometric random graph, we identify (with high probability) the rate of convergence of $$ to g in the large data limit $$. Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model.