Local Spectral Clustering of Density Upper Level Sets
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We analyze the Personalized PageRank (PPR) algorithm, a local spectral method for clustering, which extracts clusters using locally-biased random walks around a given seed node. In contrast to previous work, we adopt a classical statistical learning setup, where we obtain samples from an unknown distribution, and aim to identify connected regions of high-density (density clusters). We prove that PPR, run on a neighborhood graph, extracts sufficiently salient density clusters, that satisfy a set of natural geometric conditions. We also show a converse result, that PPR can fail to recover geometrically poorly-conditioned density clusters, even asymptotically. Finally, we provide empirical support for our theory.
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