用最优传输框架统一学习社区模型，实现参数估计与社区数自动选择。
Bridging Maximum Likelihood and Optimal Transport for Efficient Inference and Model Selection in Stochastic Block Models
- 将最大似然变分推断转化为带熵正则的半松弛格罗莫夫-沃瑟斯坦投影。
- 无正则化时可一致恢复连接矩阵和聚类分配，但有限样本下模型选择不稳定。
- 引入稀疏性约束后，单次优化即可同时完成参数估计与社区数选择。
我们从最优传输（OT）视角研究随机块模型（SBM）的推断问题。首先证明最大似然变分推断（MLVI）可视为带熵正则的半松弛格罗莫夫-沃瑟斯坦（srGW）投影。该形式虽能准确聚类，但熵正则使传输方案无法稀疏，阻碍内在模型选择。因此，我们研究无正则化的srGW估计器，证明其在渐近条件下能一致恢复SBM的连接矩阵和潜在聚类分配。然而，该渐近性质在有限样本中不保证可靠模型选择，需额外机制促进聚类比例稀疏性。实验表明，引入正则化后，估计器可在单一优化问题中同时恢复模型参数并选择社区数量，避免耗时的网格搜索或启发式方法。
原文摘要 · Abstract (English)
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.
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