  In this work, we focus on the Bipartite Stochastic Block Model (BiSBM), a
popular model for bipartite graphs with a community structure. We consider the
high dimensional setting where the number $n_1$ of type I nodes is far smaller
than the number $n_2$ of type II nodes. The recent work of Braun and Tyagi
(2022) established a sufficient and necessary condition on the sparsity level
$p_{max}$ of the bipartite graph to be able to recover the latent partition of
type I nodes. They proposed an iterative method that extends the one proposed
by Ndaoud et al. (2022) to achieve this goal. Their method requires a good
enough initialization, usually obtained by a spectral method, but empirical
results showed that the refinement algorithm doesn't improve much the
performance of the spectral method. This suggests that the spectral achieves
exact recovery in the same regime as the refinement method. We show that it is
indeed the case by providing new entrywise bounds on the eigenvectors of the
similarity matrix used by the spectral method. Our analysis extend the
framework of Lei (2019) that only applies to symmetric matrices with limited
dependencies. As an important technical step, we also derive an improved
concentration inequality for similarity matrices.
