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

We study asymptotic consistency guarantees for a non-parametric regression problem with Laplacian regularization. In particular, we consider $(x_1, y_1), \dots, (x_n, y_n)$ samples from some distribution on the cross product $\mathcal{M} \times \mathbb{R}$, where $\mathcal{M}$ is a $m$-dimensional manifold embedded in $\mathbb{R}^d$. A geometric graph on the cloud $\{x_1, \dots, x_n \}$ is constructed by connecting points that are within some specified distance $\varepsilon_n$. A suitable semi-linear equation involving the resulting graph Laplacian is used to obtain a regressor for the observed values of $y$. We establish probabilistic error rates for the uniform difference between the regressor constructed from the observed data and the Bayes regressor (or trend) associated to the ground-truth distribution. We give the explicit dependence of the rates in terms of the parameter $\varepsilon_n$, the strength of regularization $\beta_n$, and the number of data points $n$. Our argument relies on a simple, yet powerful, maximum principle for the graph Laplacian. We also address a simple extension of the framework to a semi-supervised setting.