On Mixed Memberships and Symmetric Nonnegative Matrix Factorizations

The problem of finding overlapping communities in networks has gained much attention recently. Optimization-based approaches use nonnegative matrix factorization (NMF) or variants, but the global optimum cannot be provably attained in general. Model-based approaches, such as the popular mixed membership stochastic blockmodel or MMSB (Airoldi et al., 2008), use parameters for each node to specify the overlapping communities, but standard inference techniques cannot guarantee consistency. We link the two approaches, by (a) establishing sufficient conditions for the symmetric NMF optimization to have a unique solution under MMSB, and (b) proposing a computationally efficient algorithm called GeoNMF that is provably optimal and hence consistent for a broad parameter regime. We demonstrate its accuracy on both simulated and real-world datasets.