People belong to multiple communities, words belong to multiple topics, and books cover multiple genres; overlapping clusters are commonplace. Many existing overlapping clustering methods model each person (or word, or book) as a non-negative weighted combination of "exemplars" who belong solely to one community, with some small noise. Geometrically, each person is a point on a cone whose corners are these exemplars. This basic form encompasses the widely used Mixed Membership Stochastic Blockmodel of networks and its degree-corrected variants, as well as topic models such as LDA. We show that a simple one-class SVM yields provably consistent parameter inference for all such models, and scales to large datasets. Experimental results on several simulated and real datasets show our algorithm (called SVM-cone) is both accurate and scalable.

People belong to multiple communities, words belong to multiple topics, and books cover multiple genres; overlapping clusters are commonplace. Many existing overlapping clustering methods model each person (or word, or book) as a non-negative weighted combination of "exemplars" who belong solely to one community, with some small noise. Geometrically, each person is a point on a cone whose corners are these exemplars. This basic form encompasses the widely used Mixed Membership Stochastic Blockmodel of networks (Airoldi et al., 2008) and its degree-corrected variants (Karrer et al. 2011; Jin et al., 2017), as well as topic models such as LDA (Blei et al., 2003). We show that a simple one-class SVM yields provably consistent parameter inference for all such models, and scales to large datasets. Experimental results on several simulated and real datasets show our algorithm (called SVM-cone) is both accurate and scalable.

We consider the problem of estimating overlapping community memberships. Existing provable algorithms for this problem either make strong assumptions about the population (Zhang et al., 2014; Jin et al., 2017), or are too computationally expensive (Anandkumar et al., 2014; Hopkins et al., 2017). We work under the popular Mixed Membership Stochastic Blockmodel (MMSB) (Airoldi et al., 2008). Using the inherent geometry of this model, we link the inference of overlapping communities to the problem of finding corners in a noisy rotated and scaled simplex, for which consistent algorithms exist (Gillis et al., 2008). We use this as a building block for our algorithm to infer the community memberships of each node. Furthermore, we prove that each node's soft membership vector converges to its population counterpart. To our knowledge, this is the first work to obtain rate of convergence for community membership vectors of each node, in contrast to previous work which obtain convergence results for memberships of all nodes as a whole. As a byproduct of our analysis, we derive sharp row-wise eigenvector deviation bounds, and provide a cleaning step that improves the performance significantly for sparse networks. We also propose both necessary and sufficient conditions for identifiability of the model, while existing methods typically present sufficient conditions. The empirical performance of our method is shown using simulated and real datasets scaling up to 100,000 nodes.

The problem of finding overlapping communities in networks has gained much attention recently. Optimization-based approaches use non-negative 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.