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We study the Personalized PageRank (PPR) algorithm, a local spectral methodfor clustering, which extracts clusters using locally-biased random walksaround a given seed node. In contrast to previous work, we adopt a classicalstatistical learning setup, where we obtain samples from an unknownnonparametric distribution, and aim to identify sufficiently salient clusters.We introduce a trio of population-level functionals -- the normalized cut,conductance, and local spread, analogous to graph-based functionals of the samename -- and prove that PPR, run on a neighborhood graph, recovers clusters withsmall population normalized cut and large conductance and local spread. Weapply our general theory to establish that PPR identifies connected regions ofhigh density (density clusters) that satisfy a set of natural geometricconditions. We also show a converse result, that PPR can fail to recovergeometrically poorly-conditioned density clusters, even asymptotically.Finally, we provide empirical support for our theory.

Statistical Guarantees for Local Spectral Clustering on Random Neighborhood Graphs