Finding Dense Clusters via "Low Rank + Sparse" Decomposition
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Finding "densely connected clusters" in a graph is in general an important and well studied problem in the literature Schaeffer. It has various applications in pattern recognition, social networking and data mining Duda,Mishra. Recently, Ames and Vavasis have suggested a novel method for finding cliques in a graph by using convex optimization over the adjacency matrix of the graph Ames, Ames2. Also, there has been recent advances in decomposing a given matrix into its "low rank" and "sparse" components Candes, Chandra. In this paper, inspired by these results, we view "densely connected clusters" as imperfect cliques, where imperfections correspond missing edges, which are relatively sparse. We analyze the problem in a probabilistic setting and aim to detect disjointly planted clusters. Our main result basically suggests that, one can find dense clusters in a graph, as long as the clusters are sufficiently large. We conclude by discussing possible extensions and future research directions.
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