Abstract

This paper investigates graph clustering in Th the planted cluster model in the presence le of small clusters. Traditional results dictate gr that for an algorithm to provably correctly gr recover the clusters, all clusters must be sufth ficiently large (in particular, ) where n th is the number of nodes of the graph). We ur show that this is not really a restriction: by a ne more refined analysis of the trace-norm based wo matrix recovery approach proposed in Jalali sp et al. (2011) and Chen et al. (2012), we prove ly that small clusters, under certain mild assi sumptions, do not hinder recovery of large gr ones. Based on this result, we further devise tw an iterative algorithm to recover almost all of clusters via a “peeling strategy”, i.e., recover id large clusters first, leading to a reduced problem, and repeat this procedure. These reMa sults are extended to the partial observation Co setting, in which only a (chosen) part of the pl graph is observed. The peeling strategy gives at rise to an active learning algorithm, in which nu edges adjacent to smaller clusters are queried pe more often as large clusters are learned (and ne removed). (w me From a high level, this paper sheds novel if insights on high-dimensional statistics and ?? learning structured data, by presenting a th structured matrix learning problem for which no a one shot convex relaxation approach necgr essarily fails, but a carefully constructed seti quence of convex relaxations does the job.