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

T0 review · 1 major / 2 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read Unregularized semi-relaxed Gromov-Wasserstein estimators recover stochastic block model parameters and cluster assignments in the asymptotic regime.

desk verdict The paper reinterprets SBM variational inference as semi-relaxed Gromov-Wasserstein, proves asymptotic consistency for the unregularized version, and shows empirically that regularization enables joint parameter recovery plus model selection. read the letter →

arxiv 2605.28488 v2 pith:JRAP3AVZ submitted 2026-05-27 stat.ML cs.LGmath.STstat.TH

Simon Queric , Cédric Vincent-Cuaz , Charles Bouveyron , Marco Corneli This is my paper · ORCID
classification stat.MLcs.LGmath.STstat.TH
keywords stochasticblockmodeloptimaltransportgromov-wassersteindistanceselectionvariationalinferencenetworkclusteringasymptoticconsistency
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper reinterprets maximum likelihood variational inference for stochastic block models as a semi-relaxed Gromov-Wasserstein projection under entropic regularization. It proves that removing the regularization yields consistent recovery of both the connectivity matrix and the latent cluster assignments when the number of nodes tends to infinity. Because this consistency does not guarantee good model selection in finite samples, the authors add a regularization term that promotes sparsity in the cluster proportions, allowing the same optimization to recover parameters and select the number of clusters at once.

What carries the argument

semi-relaxed Gromov-Wasserstein projection, which reformulates the variational inference objective and permits direct analysis of consistency for recovering SBM parameters.

What would settle it

A finite-sample simulation in which the regularized srGW estimator selects a number of clusters different from the true value would challenge the claim of simultaneous reliable selection.

Watch

Extended reading notes

Core claim

Maximum likelihood variational inference in stochastic block models is equivalent to a semi-relaxed Gromov-Wasserstein projection with entropic regularization. Unregularized srGW estimators consistently recover the SBM connectivity matrix and latent cluster assignments asymptotically. A regularized formulation performs both parameter estimation and selection of the number of clusters simultaneously in a single optimization problem.

Load-bearing premise

That sparsity-promoting regularization can be introduced to achieve reliable model selection in finite samples while preserving the recovery properties.

Editorial extensions

If this is right

  • Consistent asymptotic recovery of the connectivity matrix and assignments.
  • Joint inference and model selection avoids separate grid search procedures.
  • Regularization addresses the sparsity issue that prevents reliable finite-sample selection.
  • Provides an efficient alternative to traditional heuristic model selection in SBMs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The connection between MLVI and srGW may allow importing other optimal transport techniques to improve SBM inference.
  • Similar reformulations could apply to other network models beyond SBMs.
  • Finite-sample performance might improve further with adaptive regularization parameters.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 2 minor

Summary. The manuscript interprets maximum likelihood variational inference for stochastic block models as an entropically regularized semi-relaxed Gromov-Wasserstein projection. It proves that unregularized srGW estimators are consistent for recovering the connectivity matrix and latent assignments in the asymptotic regime, while acknowledging that this consistency does not yield reliable finite-sample model selection. It then presents an empirical demonstration that an additional regularized formulation performs joint parameter recovery and selection of the number of clusters within a single optimization problem.

Significance. If the consistency result is rigorously established, the work supplies a useful OT-based reinterpretation of MLVI for SBMs and a practical route to model selection that avoids separate grid search. The explicit recognition that asymptotic consistency alone is insufficient for finite-sample selection is a positive feature; the empirical evidence for the regularized joint estimator would need to be robust across multiple regimes to support the efficiency claim.

major comments (1)
  1. [Abstract] Abstract (final paragraph): the central claim that the regularized formulation 'yields estimators that simultaneously recover model parameters and select the number of clusters in a single optimization problem' rests entirely on empirical demonstration; because the manuscript does not supply a corresponding consistency or selection-consistency theorem for the regularized case, the load-bearing step from asymptotic unregularized recovery to finite-sample joint selection requires stronger empirical controls (e.g., explicit comparison against standard criteria such as BIC or ICL on the same simulated and real networks).
minor comments (2)
  1. The notation for the semi-relaxed Gromov-Wasserstein objective and its relation to the SBM likelihood should be introduced with an explicit equation linking the two formulations (currently only described in prose).
  2. Figure captions and experimental details should state the precise form of the sparsity-promoting regularization (e.g., the value of any additional penalty parameter) and the range of network sizes used to illustrate finite-sample behavior.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final paragraph): the central claim that the regularized formulation 'yields estimators that simultaneously recover model parameters and select the number of clusters in a single optimization problem' rests entirely on empirical demonstration; because the manuscript does not supply a corresponding consistency or selection-consistency theorem for the regularized case, the load-bearing step from asymptotic unregularized recovery to finite-sample joint selection requires stronger empirical controls (e.g., explicit comparison against standard criteria such as BIC or ICL on the same simulated and real networks).

    Authors: We agree that the central claim for the regularized formulation is supported by empirical evidence rather than a consistency theorem, as already noted in the manuscript. To strengthen the empirical validation, we will add explicit comparisons of the regularized srGW estimator against BIC and ICL on the same simulated and real networks in the revised version. This will provide more robust controls for the joint recovery and selection performance. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation chain is self-contained

full rationale

The abstract presents an interpretation of MLVI as entropically regularized srGW, followed by a consistency proof for the unregularized case in the asymptotic regime and an empirical demonstration for the regularized formulation's joint parameter recovery and model selection. No quoted equations or steps reduce a claimed prediction or result to a fitted quantity defined in terms of itself, nor do they rely on self-citation chains or imported uniqueness theorems. The paper explicitly flags the gap between asymptotic consistency and finite-sample reliability, treating regularization as an empirical mechanism rather than a derived necessity. This structure keeps the central claims independent of the inputs they are derived from.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; full manuscript would be required to audit fitted scales, background assumptions, or new constructs.

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Cite this review

Pith. "Pith review of Bridging Maximum Likelihood and Optimal Transport for Efficient Inference and Model Selection in Stochastic Block Models." pith.science (2026). https://pith.science/paper/JRAP3AVZ

@misc{pith2026260528488,
  author       = {Pith},
  title        = {Pith review of: Bridging Maximum Likelihood and Optimal Transport for Efficient Inference and Model Selection in Stochastic Block Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JRAP3AVZ}},
  note         = {Machine review of arXiv:2605.28488}
}
read the original abstract

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.

Figures

Figures reproduced from arXiv: 2605.28488 by the authors.

Figure 1
Figure 1. Left : A graph sampled from a Bernoulli SBM with five communities. Right : The [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Evolution of the average ARI w.r.t α, 5 simulated graphs for each α. The higher α, the easier to detect the clusters. 3 2 1 0 log10 ( ) 0 5 10 15 20 Estimated K Assortative =0.03 =0.06 =0.10 =0.13 =0.17 =0.20 True K 3 2 1 0 log10 ( ) 0 5 10 15 20 Hub 3 2 1 0 log10 ( ) 0 5 10 15 20 Disassortative [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Average selected K∗ versus (log of) the sparsity hyper-parameter λ. widely used community detection methods respectively based on modularity maximization and flow compression. We also consider two model-based competitors: Greed [20], which performs greedy maximization of the Integrated Classification Likelihood (ICL), and the blockmodels package [57], which implements variational EM with ICL-based model selection. E… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Tree structure of our results up to Theorem 1. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Sampled Adjacency matrices for the 3 SBM scenarions described in Section 5, i.e. [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: Evolution of ARI wrt α, the greater α, the easier to detect the clusters. We take a graph with N = 103 nodes. β = 0.03 and α ∈ [β, 0.2]. Each algorithm, excepts Louvain is searching for at most K = 20 clusters, the real number of cluster is K∗ = 5. Here the classes are…
Figure 7
Figure 7. Figure 7: Running times for different graph clustering methods. [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Running Times of srGW NLL on cpu vs gpu. [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]

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    the alternative version is true for all SBMs whereas the original only holds for models which satisfyE[A ij|Z] = Θ ZiZj (the Bernoulli model is a specific case here)

  56. [64]

    projection

    the proof of proposition of the alternative version exploits the KL divergence, whereas the one of the original version exploits the definition Bregman divergence. Moreover, a summary of what proved as preliminar to the statement of Theorem 1 cans be seen in Figure 4. 2Recall ...

  57. [65]

    Justification: The research does not involve human subjects, therefore IRB approval is not applicable

    Institutional review board (IRB) approvals or equivalent for research with human subjects Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals...

    2025

Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.