  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.
