Provable Symmetric Nonnegative Matrix Factorization for Overlapping Clustering

The problem of finding overlapping communities in networks has gained much attention recently. Algorithmic approaches often employ non-negative matrix factorization (NMF) or variants, while model-based approaches (such as the widely used mixed-membership stochastic blockmodel, or MMSB) assume a distribution over communities for each node and run standard inference techniques to recover these parameters. However, few of these approaches have provable consistency guarantees. We investigate the use of the symmetric NMF (or SNMF) for the MMSB model, and provide conditions under which an optimal SNMF algorithm can recover the MMSB parameters consistently. Since we are unaware of general-purpose optimal SNMF algorithms, we develop an SNMF variant, called GeoNMF, designed specifically for the MMSB model. GeoNMF is provably consistent, and experiments on both simulated and real-world datasets show its accuracy.