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Rates of convergence for regression with the graph poly-Laplacian.

Nicolás García Trillos1, Ryan Murray2, Matthew Thorpe3

  • 1Department of Statistics, University of Wisconsin-Madison, Madison, WI 53706 USA.

Sampling Theory, Signal Processing, and Data Analysis
|December 1, 2023
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Summary

This study introduces graph poly-Laplacian regularization for non-parametric regression. It identifies the convergence rate of the smoothed function to the true function in a large data limit, showing promising results comparable to traditional smoothing splines.

Keywords:
Asymptotic consistencyNon-parametric regression on unknown domainsNonlocal variational problemsPDEs on graphsRates of convergenceSupervised learning

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Area of Science:

  • Machine Learning
  • Graph Signal Processing
  • Statistical Learning Theory

Background:

  • Smoothing splines are a standard tool for non-parametric regression, utilizing Laplacian regularization for smoothness.
  • Higher-order regularity can be achieved using poly-Laplacian regularization, extending the capabilities of traditional methods.
  • Adapting these regularization techniques to graph-structured data is crucial for analyzing complex networks.

Purpose of the Study:

  • To investigate the application of graph poly-Laplacian regularization in a fully supervised, non-parametric regression setting.
  • To analyze the convergence rate of the proposed method for noisy graph data in the large data limit.
  • To compare the performance of graph poly-Laplacian regularization against the standard smoothing spline model.

Main Methods:

  • Formulating a variational problem minimizing an energy function composed of data fidelity and graph poly-Laplacian terms.
  • Considering a dataset with noisy labels and applying the method to a geometric random graph.
  • Analyzing the convergence rate of the minimizer to the true underlying function under independent and identically distributed (i.i.d.) noise assumptions.

Main Results:

  • The study identifies, with high probability, the rate of convergence of the estimated function to the true function.
  • This convergence rate is established in the large data limit (N → ∞).
  • The derived convergence rate is shown to be comparable to the known rate for the standard smoothing spline model.

Conclusions:

  • Graph poly-Laplacian regularization is a viable and effective method for non-parametric regression on graph-structured data.
  • The method provides theoretical guarantees on convergence rates, especially in the presence of noise and large datasets.
  • The findings suggest that graph poly-Laplacian regularization offers a powerful extension of smoothing splines for graph data analysis.