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