Rates of Convergence for Regression with the Graph Poly-Laplacian

TL;DR

This paper analyzes the convergence rates of graph poly-Laplacian regularization in non-parametric regression, showing that it achieves rates comparable to classical smoothing splines in large data limits.

Contribution

It extends convergence rate analysis to graph poly-Laplacian regularization, bridging graph-based methods with classical smoothing spline theory.

Findings

Convergence rates match those of classical smoothing splines up to logarithmic factors.

High probability bounds are established for the rate of convergence.

Results apply to geometric random graphs in a supervised regression setting.

Abstract

In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularisation. 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 regularisation in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset $\{x_i\}_{i=1}^n$ and a set of noisy labels $\{y_i\}_{i=1}^n\subset\mathbb{R}$ we let $u_n:\{x_i\}_{i=1}^n\to\mathbb{R}$ be the minimiser of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When $y_i = g(x_i)+\xi_i$, 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, up to logarithms, coincides with the known rate of convergence in the usual smoothing spline model.