# A maximum principle argument for the uniform convergence of graph Laplacian regressors

## TL;DR

This paper introduces maximum principle techniques from PDEs to prove the uniform convergence of graph Laplacian regressors in noisy, manifold-based non-parametric regression, providing new error estimates and insights.

## Contribution

It develops a novel maximum principle approach for analyzing graph Laplacian regressors, enhancing theoretical understanding and parameter selection in manifold learning.

## Key findings

- Establishes asymptotic consistency of graph Laplacian regressors.
- Provides concrete error estimates for parameter tuning.
- Connects Laplacian methods with kernel and k-NN techniques.

## Abstract

This paper investigates the use of methods from partial differential equations and the Calculus of variations to study learning problems that are regularized using graph Laplacians. Graph Laplacians are a powerful, flexible method for capturing local and global geometry in many classes of learning problems, and the techniques developed in this paper help to broaden the methodology of studying such problems. In particular, we develop the use of maximum principle arguments to establish asymptotic consistency guarantees within the context of noise corrupted, non-parametric regression with samples living on an unknown manifold embedded in $\mathbb{R}^d$. The maximum principle arguments provide a new technical tool which informs parameter selection by giving concrete error estimates in terms of various regularization parameters. A review of learning algorithms which utilize graph Laplacians, as well as previous developments in the use of differential equation and variational techniques to study those algorithms, is given. In addition, new connections are drawn between Laplacian methods and other machine learning techniques, such as kernel regression and k-nearest neighbor methods.