Rates of Convergence for Regression with the Graph Poly-Laplacian

• URL: http://arxiv.org/abs/2209.02305v1
• Date: Tue, 6 Sep 2022 08:59:15 GMT
• Title: Rates of Convergence for Regression with the Graph Poly-Laplacian
• Authors: Nicol\'as Garc\'ia Trillos, Ryan Murray, Matthew Thorpe
• Abstract summary: Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. We consider graph poly-Laplacian regularisation in a fully supervised, non-parametric, noise corrupted, regression problem.
• Score: 3.222802562733786
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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