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

arXiv: https://arxiv.org/abs/1901.10089 · HTML: https://mlsanity.com/paper/1901.10089/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2019.

## Built on by (most cited first, 9 of 9)

- [Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphs](https://mlsanity.com/paper/2106.01529/) (2021)
- [Uniform Convergence Rates for Lipschitz Learning on Graphs](https://mlsanity.com/paper/2111.12370/) (2021)
- [Minimax Optimal Regression over Sobolev Spaces via Laplacian Eigenmaps on Neighborhood Graphs](https://mlsanity.com/paper/2111.07394/) (2021)
- [Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression](https://mlsanity.com/paper/2311.00140/) (2023)
- [Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications](https://mlsanity.com/paper/2111.03217/) (2021)
- [Rates of Convergence for Regression with the Graph Poly-Laplacian](https://mlsanity.com/paper/2209.02305/) (2022)
- [Eikonal depth: an optimal control approach to statistical depths](https://mlsanity.com/paper/2201.05274/) (2022)
- [A Unified Analysis of Multi-task Functional Linear Regression Models with Manifold Constraint and Composite Quadratic Penalty](https://mlsanity.com/paper/2211.04874/) (2022)
- [Planner-Admissible Graph-PDE Value Extensions for Sparse Goal-Conditioned Planning](https://mlsanity.com/paper/2605.19185/) (2026)

## Abstract (arXiv)

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 ℝd. 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.