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Title: 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$$\{x_i\}_{i=1}^n$${xi}i=1nand a set of noisy labels$$\{y_i\}_{i=1}^n\subset \mathbb {R}$${yi}i=1nRwe let$$u_n{:}\{x_i\}_{i=1}^n\rightarrow \mathbb {R}$$un:{xi}i=1nRbe the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When$$y_i = g(x_i)+\xi _i$$yi=g(xi)+ξi, for iid noise$$\xi _i$$ξi, and using the geometric random graph, we identify (with high probability) the rate of convergence of$$u_n$$untogin 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.

 
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Award ID(s):
2005797
NSF-PAR ID:
10475993
Author(s) / Creator(s):
; ;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Sampling Theory, Signal Processing, and Data Analysis
Volume:
21
Issue:
2
ISSN:
2730-5716
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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