  The 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 and most results are limited to the binary setting. We prove here
theoretical guarantees in the valued settings. We show 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 infinity, (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, and under mild conditions, asymptotic
normality, to the maximum likelihood estimate under the observed model.
Moreover, the variational estimator is also consistent and, under the same
conditions, asymptotically normal.
