Improved clustering algorithms for the Bipartite Stochastic Block Model

TL;DR

This paper introduces a new, computationally simple algorithm for exact node partition recovery in Bipartite Stochastic Block Models, improving theoretical recovery conditions and demonstrating fast, near-optimal performance through numerical experiments.

Contribution

It proposes a novel spectral initialization and Lloyd's iteration-based procedure that achieves exact recovery under milder conditions than existing methods in BSBM.

Findings

The new algorithm achieves exact recovery under the condition p = Ω(max(√(log n₁)/(n₁ n₂), log n₁/n₂)).

Numerical results show the algorithm is fast and nearly matches the performance of an oracle procedure.

The method improves the known bounds for the number of clauses needed for complete recovery in related satisfiability problems.

Abstract

We establish sufficient conditions of exact and almost full recovery of the node partition in Bipartite Stochastic Block Model (BSBM) using polynomial time algorithms. First, we improve upon the known conditions of almost full recovery by spectral clustering algorithms in BSBM. Next, we propose a new computationally simple and fast procedure achieving exact recovery under milder conditions than the state of the art. Namely, if the vertex sets $V_1$ and $V_2$ in BSBM have sizes $n_1$ and $n_2$, we show that the condition $p = \Omega\left(\max\left(\sqrt{\frac{\log{n_1}}{n_1n_2}},\frac{\log{n_1}}{n_2}\right)\right)$ on the edge intensity $p$ is sufficient for exact recovery witin $V_1$. This condition exhibits an elbow at $n_{2} \asymp n_1\log{n_1}$ between the low-dimensional and high-dimensional regimes. The suggested procedure is a variant of Lloyd's iterations initialized with a well-chosen spectral estimator leading to what we expect to be the optimal condition for exact recovery in BSBM. {The optimality conjecture is supported by showing that, for a supervised oracle procedure, such a condition is necessary to achieve exact recovery.} The key elements of the proof techniques are different from classical community detection tools on random graphs. Numerical studies confirm our theory, and show that the suggested algorithm is both very fast and achieves {almost the same} performance as the supervised oracle. 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.