Similarity graphs provide a structure for analyzing high dimensional data. These undirected weighted graphs provide structure for identifying inherent clusters in datasets and many methods exist to sort through such data building on the graph laplacian matrix. One way to think about such problems is in terms of penalized cut problems. These can be expressed in terms of the graph total variation which has a well-known analogue in Euclidean space. We show how to use ideas from geometric methods for PDEs to develop efficient and high performing methods for semi-supervised and unsupervised learning. These methods also extend to active learning and to modularity optimization for community detection on networks.