  Graph-based methods have been quite successful in solving unsupervised and
semi-supervised learning problems, as they provide a means to capture the
underlying geometry of the dataset. It is often desirable for the constructed
graph to satisfy two properties: first, data points that are similar in the
feature space should be strongly connected on the graph, and second, the class
label information should vary smoothly with respect to the graph, where
smoothness is measured using the spectral properties of the graph Laplacian
matrix. Recent works have justified some of these smoothness conditions by
showing that they are strongly linked to the semi-supervised smoothness
assumption and its variants. In this work, we reinforce this connection by
viewing the problem from a graph sampling theoretic perspective, where class
indicator functions are treated as bandlimited graph signals (in the
eigenvector basis of the graph Laplacian) and label prediction as a bandlimited
reconstruction problem. Our approach involves analyzing the bandwidth of class
indicator signals generated from statistical data models with separable and
nonseparable classes. These models are quite general and mimic the nature of
most real-world datasets. Our results show that in the asymptotic limit, the
bandwidth of any class indicator is also closely related to the geometry of the
dataset. This allows one to theoretically justify the assumption of
bandlimitedness of class indicator signals, thereby providing a sampling
theoretic interpretation of graph-based semi-supervised classification.
