Semi-supervised Learning using Sparse Eigenfunction Bases

We present a new framework for semi-supervised learning with sparse eigenfunction bases of kernel matrices. It turns out that when the \emph{cluster assumption} holds, that is, when the high density regions are sufficiently separated by low density valleys, each high density area corresponds to a unique representative eigenvector. Linear combination of such eigenvectors (or, more precisely, of their Nystrom extensions) provide good candidates for good classification functions. By first choosing an appropriate basis of these eigenvectors from unlabeled data and then using labeled data with Lasso to select a classifier in the span of these eigenvectors, we obtain a classifier, which has a very sparse representation in this basis. Importantly, the sparsity appears naturally from the cluster assumption.