Spectral thresholds in the bipartite stochastic block model
29th Annual Conference on Learning Theory, PMLR 49:943-959, 2016.
We consider a bipartite stochastic block model on vertex sets V_1 and V_2, with planted partitions in each, and ask at what densities efficient algorithms can recover the partition of the smaller vertex set. When |V_2| ≫|V_1|, multiple thresholds emerge. We first locate a sharp threshold for detection of the partition, in the sense of the results of Mossel, Neeman and Sly and Massoulie for the stochastic block model. We then show that at a higher edge density, the singular vectors of the rectangular biadjacency matrix exhibit a localization / delocalization phase transition, giving recovery above the threshold and no recovery below. Nevertheless, we propose a simple spectral algorithm, Diagonal Deletion SVD, which recovers the partition at a nearly optimal edge density. The bipartite stochastic block model studied here was used by Feldman, Perkins, Vempala to give a unified algorithm for recovering planted partitions and assignments in random hypergraphs and random k-SAT formulae respectively. Our results give the best known bounds for the clause density at which solutions can be found efficiently in these models as well as showing a barrier to further improvement via this reduction to the bipartite block model.