On the Consistency of Graph-based Bayesian Learning and the Scalability of Sampling Algorithms

A popular approach to semi-supervised learning proceeds by endowing the inputdata with a graph structure in order to extract geometric information and incorporate it intoa Bayesian framework. We introduce new theory that gives appropriate scalings of graphparameters that provably lead to a well-defined limiting posterior as the size of the unlabeleddata set grows. Furthermore, we show that these consistency results have profound algorithmicimplications. When consistency holds, carefully designed graph-based Markov chain MonteCarlo algorithms are proved to have a uniform spectral gap, independent of the number ofunlabeled inputs. Several numerical experiments corroborate both the statistical consistencyand the algorithmic scalability established by the theory.