Title: Consistency and Asymptotic Normality of Latent Blocks Model Estimators,
Abstract: Latent Block Model (LBM) is a model-based method to cluster simultaneously
the $d$ columns and $n$ rows of a data matrix. Parameter estimation in LBM is a
difficult and multifaceted problem. Although various estimation strategies have
been proposed and are now well understood empirically, theoretical guarantees
about their asymptotic behavior is rather sparse. We show here that under some
mild conditions on the parameter space, and in an asymptotic regime where
$\log(d)/n$ and $\log(n)/d$ tend to $0$ when $n$ and $d$ tend to $+\infty$, (1)
the maximum-likelihood estimate of the complete model (with known labels) is
consistent and (2) the log-likelihood ratios are equivalent under the complete
and observed (with unknown labels) models. This equivalence allows us to
transfer the asymptotic consistency to the maximum likelihood estimate under
the observed model. Moreover, the variational estimator is also consistent.