Convergence of the groups posterior distribution in latent or stochastic block models

TL;DR

This paper introduces a unified framework for analyzing the convergence of the groups posterior distribution in latent and stochastic block models, providing conditions for asymptotic group recovery and discussing model limitations.

Contribution

It offers a comprehensive theoretical analysis of the groups posterior convergence in both latent and stochastic block models, including conditions for successful recovery.

Findings

Conditions for groups posterior to converge to true groups

Identification of symmetry cases preventing recovery

Analysis of model behavior when data sparsity increases

Abstract

We propose a unified framework for studying both latent and stochastic block models, which are used to cluster simultaneously rows and columns of a data matrix. In this new framework, we study the behaviour of the groups posterior distribution, given the data. We characterize whether it is possible to asymptotically recover the actual groups on the rows and columns of the matrix, relying on a consistent estimate of the parameter. In other words, we establish sufficient conditions for the groups posterior distribution to converge (as the size of the data increases) to a Dirac mass located at the actual (random) groups configuration. In particular, we highlight some cases where the model assumes symmetries in the matrix of connection probabilities that prevents recovering the original groups. We also discuss the validity of these results when the proportion of non-null entries in the data matrix converges to zero.