# Bridging Maximum Likelihood and Optimal Transport for Efficient Inference and Model Selection in Stochastic Block Models

arXiv: https://arxiv.org/abs/2605.28488 · HTML: https://mlsanity.com/paper/2605.28488/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 63 references that are in the map.
- First posted on arXiv 27 May 2026.

## Builds on (most cited first, 27 of 27)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Optimal Transport for structured data with application on graphs](https://mlsanity.com/paper/1805.09114/) (2018)
- [Consistency of spectral clustering in stochastic block models](https://mlsanity.com/paper/1312.2050/) (2013)
- [Gromov-Wasserstein Learning for Graph Matching and Node Embedding](https://mlsanity.com/paper/1901.06003/) (2019)
- [Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching](https://mlsanity.com/paper/1905.07645/) (2019)
- [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017)
- [A Review of Stochastic Block Models and Extensions for Graph Clustering](https://mlsanity.com/paper/1903.00114/) (2019)
- [Generalized Spectral Clustering via Gromov-Wasserstein Learning](https://mlsanity.com/paper/2006.04163/) (2020)
- [Learning Graphons via Structured Gromov-Wasserstein Barycenters](https://mlsanity.com/paper/2012.05644/) (2020)
- [Online Graph Dictionary Learning](https://mlsanity.com/paper/2102.06555/) (2021)
- [A Survey on Optimal Transport for Machine Learning: Theory and Applications](https://mlsanity.com/paper/2106.01963/) (2021)
- [Heat kernel based community detection](https://mlsanity.com/paper/1403.3148/) (2014)
- [Adjusting for Chance Clustering Comparison Measures](https://mlsanity.com/paper/1512.01286/) (2015)
- [Gromov-Wasserstein Factorization Models for Graph Clustering](https://mlsanity.com/paper/1911.08530/) (2019)
- [Spectral Clustering of Graphs with the Bethe Hessian](https://mlsanity.com/paper/1406.1880/) (2014)
- [Semi-relaxed Gromov-Wasserstein divergence with applications on graphs](https://mlsanity.com/paper/2110.02753/) (2021)
- [Template based Graph Neural Network with Optimal Transport Distances](https://mlsanity.com/paper/2205.15733/) (2022)
- [Quantized Gromov-Wasserstein](https://mlsanity.com/paper/2104.02013/) (2021)
- [Mean-field theory of graph neural networks in graph partitioning](https://mlsanity.com/paper/1810.11908/) (2018)
- [A Gromov--Wasserstein Geometric View of Spectrum-Preserving Graph Coarsening](https://mlsanity.com/paper/2306.08854/) (2023)
- [Recovering communities in the general stochastic block model without knowing the parameters](https://mlsanity.com/paper/1506.03729/) (2015)
- [Generalized Dimension Reduction Using Semi-Relaxed Gromov-Wasserstein Distance](https://mlsanity.com/paper/2405.15959/) (2024)
- [Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein](https://mlsanity.com/paper/2402.02239/) (2024)
- [The quest for the GRAph Level autoEncoder (GRALE)](https://mlsanity.com/paper/2505.22109/) (2025)
- [NeuroCUT: A Neural Approach for Robust Graph Partitioning](https://mlsanity.com/paper/2310.11787/) (2023)
- [A note on the relations between mixture models, maximum-likelihood and entropic optimal transport](https://mlsanity.com/paper/2501.12005/) (2025)
- [Optimal Transport-Based Clustering of Attributed Graphs with an Application to Road Traffic Data](https://mlsanity.com/paper/2512.15570/) (2025)

## Abstract (arXiv)

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.