  Graph-based semi-supervised learning is the problem of propagating labels
from a small number of labelled data points to a larger set of unlabelled data.
This paper is concerned with the consistency of optimization-based techniques
for such problems, in the limit where the labels have small noise and the
underlying unlabelled data is well clustered. We study graph-based probit for
binary classification, and a natural generalization of this method to
multi-class classification using one-hot encoding. The resulting objective
function to be optimized comprises the sum of a quadratic form defined through
a rational function of the graph Laplacian, involving only the unlabelled data,
and a fidelity term involving only the labelled data. The consistency analysis
sheds light on the choice of the rational function defining the optimization.
