AI 中文总结
本文证明谱初始化变分EM在一般随机块模型中达到Chernoff指数级最优社区恢复，无需全局最大化，且实现一阶精确恢复阈值。
AI 中文摘要
我们证明了在规定的迭代预算内，谱初始化的批量变分EM输出的Chernoff指数保证，而无需假设其变分目标达到全局最大化。该迭代从同一稀疏图中反复估计整个块概率矩阵和社区比例。我们考虑固定数量的社区，其比例远离零，且具有固定、正且不同的连接性分布；既不要求同配性，也不要求满秩。该算法使用同步softmax更新，无需样本分割或后验阈值化。其分析必须在指数级小的风险尺度上控制估计参数、软标签和重用边之间的反馈。我们在数据相关的软分配上建立了一个均匀的一步界，其随机余数的期望呈指数级小。结合指数级可靠的正则化谱初始化器，这在整个稀疏、发散度区域中产生了由$\exp\{-(1-o(1))J_n\}$界定的无条件期望误分类率，其中$J_n$是逐节点的最小Chernoff信息。匹配的下界确立了在允许未知连接性和不同社区数量的局部参数空间上的一阶对数极小极大最优性。该算法还达到了尖锐的一阶精确恢复阈值。数值实验展示了细化增益以及对初始化和不平衡性的敏感性。
英文摘要
We prove a Chernoff-exponent guarantee for the output of spectrally initialized batch variational EM after a prescribed iteration budget, without assuming global maximization of its variational objective. The iteration repeatedly estimates the entire block probability matrix and community proportions from the same sparse graph. We consider a fixed number of communities with proportions bounded away from zero and fixed, positive, distinct connectivity profiles; neither assortativity nor full rank is required. The algorithm uses simultaneous softmax updates without sample splitting or posterior thresholding. Its analysis must control the feedback between estimated parameters, soft labels and reused edges at an exponentially small risk scale. We establish a uniform one-step bound over data-dependent soft assignments whose random remainder has exponentially small expectation. Combined with an exponentially reliable regularized spectral initializer, this yields an unconditional expected misclassification rate bounded by $\exp\{-(1-o(1))J_n\}$ throughout the sparse, diverging-degree regime, where $J_n$ is the minimum nodewise Chernoff information. Matching lower bounds establish first-order logarithmic minimax optimality on local parameter spaces allowing unknown connectivity and varying community counts. The algorithm also attains the sharp first-order exact-recovery threshold. Numerical experiments illustrate refinement gains and sensitivity to initialization and imbalance.
Comments23 pages, 4 figures. Main text only; supplementary material is not included
