Title: On the Consistency of Graph-based Bayesian Learning and the Scalability of Sampling Algorithms,
Abstract: A popular approach to semi-supervised learning proceeds by endowing the input
data with a graph structure in order to extract geometric information and
incorporate it into a Bayesian framework. We introduce new theory that gives
appropriate scalings of graph parameters that provably lead to a well-defined
limiting posterior as the size of the unlabeled data set grows. Furthermore, we
show that these consistency results have profound algorithmic implications.
When consistency holds, carefully designed graph-based Markov chain Monte Carlo
algorithms are proved to have a uniform spectral gap, independent of the number
of unlabeled inputs. Several numerical experiments corroborate both the
statistical consistency and the algorithmic scalability established by the
theory.