Improved clustering algorithms for the Bipartite Stochastic Block Model
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We consider a Bipartite Stochastic Block Model (BSBM) on vertex sets V_1 and V_2, and investigate asymptotic sufficient conditions of exact and almost full recovery for polynomial-time algorithms of clustering over V_1, in the regime where the cardinalities satisfy |V_1|≪|V_2|. We improve upon the known conditions of almost full recovery for spectral clustering algorithms in BSBM. Furthermore, we propose a new computationally simple procedure achieving exact recovery under milder conditions than the state of the art. This procedure is a variant of Lloyd's iterations initialized with a well-chosen spectral algorithm leading to what we expect to be optimal conditions for exact recovery in this model. The key elements of the proof techniques are different from classical community detection tools on random graphs. In particular, we develop a heavy-tailed variant of matrix Bernstein inequality. Finally, using the connection between planted satisfiability problems and the BSBM, we improve upon the sufficient number of clauses to completely recover the planted assignment.
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