Rates of convergence for regression with the graph poly-Laplacian

Abstract

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