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Articles (2021), IEEE Transactions on Information Theory, In press

Improved clustering algorithms for the Bipartite Stochastic Block Model

NDAOUD Mohamed , SIGALA Suzanne, TSYBAKOV Alexandre

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 = Omegaleft(maxleft(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 $n_{2} asymp n_1log{n_1}$ 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. Lien vers l'article

NDAOUD, M., SIGALA, S. and TSYBAKOV, A. (2021). Improved clustering algorithms for the Bipartite Stochastic Block Model. IEEE Transactions on Information Theory, In press.

Mots clés : #Bipartite-Stochastic-Block-Model, #exact-recovery, #almost-full-recovery, #spectral-methods, #clustering, #phase-transition