Bridging Maximum Likelihood and Optimal Transport for Efficient Inference and Model Selection in Stochastic Block Models
For practitioners using stochastic block models, this work provides a principled method for joint inference and model selection without costly hyperparameter tuning.
The paper connects maximum likelihood variational inference in stochastic block models to optimal transport, showing that unregularized estimators recover parameters asymptotically but require sparsity regularization for finite-sample model selection. The proposed regularized formulation simultaneously estimates parameters and selects the number of clusters in a single optimization, avoiding grid search.
We study inference in stochastic block models (SBMs) through the lens of optimal transport (OT). We first establish that maximum likelihood variational inference (MLVI) can be interpreted as a semi-relaxed Gromov-Wasserstein (srGW) projection with entropic regularization. While this formulation yields accurate clustering, the entropic regularization prevents transport plans to be sparse, hindering intrinsic model selection. Consequently, we investigate unregularized srGW estimators, and prove that they consistently recover both the SBM connectivity matrix and latent cluster assignments in the asymptotic regime. However, this asymptotic property does not translate into reliable model selection in finite samples, and calls for additional mechanisms to promote sparsity in the inferred cluster proportions. We empirically show that such a regularized formulation yields estimators that simultaneously recover model parameters and select the number of clusters in a single optimization problem, thereby avoiding costly grid search or heuristic model selection procedures.