Tikhonov regularization and semi-supervised learning on large graphs

We consider the problem of labeling a partially labeled graph. This setting may arise in a number of situations from survey sampling to information retrieval to pattern recognition in manifold settings. It is also, especially, of potential practical importance when data is abundant, but labeling is expensive or requires human assistance. Our approach develops a framework for regularization on such graphs parallel to Tikhonov regularization on continuous spaces. The algorithms are very simple and involve solving a single, usually sparse, system of linear equations. Using the notion of algorithmic stability, we derive bounds on the generalization error and relate it to the structural invariants of the graph.