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

Abstract

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 ${\left\{x_i\right\}}_{i=1}^n$ and a set of noisy labels ${\left\{y_i\right\}}_{i=1}^n\subset R$ we let $u_n:{\left\{x_i\right\}}_{i=1}^n\to R$ be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When $y_i=g\left(x_i\right)+{\xi }_i$, for iid noise ${\xi }_i$, and using the geometric random graph, we identify (with high probability) the rate of convergence of $u_n$ to g in the large data limit $n\to \infty$. Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model.