Semi-supervised learning with the graph laplacian: The limit of infinite unlabelled data

Abstract

We study the behavior of the popular Laplacian Regularization method for Semi-Supervised Learning at the regime of a fixed number of labeled points but a large number of unlabeled points. We show that in ℝd, d ≥ 2, the method is actually not well-posed, and as the number of unlabeled points increases the solution degenerates to a noninformative function. We also contrast the method with the Laplacian Eigenvector method, and discuss the "smoothness" assumptions associated with this alternate method.