Finding Dense Clusters via "Low Rank + Sparse" Decomposition
Abstract
Finding "densely connected clusters" in a graph is in general an important and well studied problem in the literature \cite{Schaeffer}. It has various applications in pattern recognition, social networking and data mining \cite{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 \cite{Ames, Ames2}. Also, there has been recent advances in decomposing a given matrix into its "low rank" and "sparse" components \cite{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 \emph{dense} clusters in a graph, as long as the clusters are sufficiently large. We conclude by discussing possible extensions and future research directions.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2011
- DOI:
- 10.48550/arXiv.1104.5186
- arXiv:
- arXiv:1104.5186
- Bibcode:
- 2011arXiv1104.5186O
- Keywords:
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- Statistics - Machine Learning;
- Computer Science - Information Theory
- E-Print:
- 19 pages, 2 figures