Eigenvector localization on data-dependent graphs

We aim to understand and characterize embeddings of datasets with small anomalous clusters using the Laplacian Eigenmaps algorithm. To do this, we characterize the order in which eigenvectors of a disjoint graph Laplacian emerge and the support of those eigenvectors. We then extend this characterization to weakly connected graphs with clusters of differing sizes, utilizing the theory of invariant subspace perturbations and proving some novel results. Finally, we propose a simple segmentation algorithm for anomalous clusters based off our theory.