Rates of Convergence for Regression with the Graph Poly-Laplacian
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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⊂ℝ we let u_n:{x_i}_i=1^n→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)+ξ_i, for iid noise ξ_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→∞. Furthermore, our rate, up to logarithms, coincides with the known rate of convergence in the usual smoothing spline model.
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