# Consistency and Asymptotic Normality of Latent Block Model Estimators

## TL;DR

This paper establishes theoretical guarantees for the consistency and asymptotic normality of estimators in the Latent Block Model, extending results to valued data settings and under mild asymptotic conditions.

## Contribution

It provides the first theoretical proof of consistency and asymptotic normality for estimators in valued Latent Block Models, including variational estimators.

## Key findings

- MLE is consistent with known labels under mild conditions.
- Log-likelihood ratios are equivalent between complete and observed models.
- Variational estimators are also consistent and asymptotically normal.

## Abstract

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.