标题： Softmax门控高斯专家混合模型的混合度树状图：无需模型扫描的一致性 显示英文标题

标题： Dendrograms of Mixing Measures for Softmax-Gated Gaussian Mixture of Experts: Consistency without Model Sweeps

摘要： 我们开发了一个统一的统计框架，用于softmax门控高斯专家混合模型（SGMoE），解决了参数估计和模型选择中的三个长期障碍：(i) 门控参数在共同平移下的不可识别性，(ii) 内在的门控-专家相互作用导致似然中的耦合微分关系，以及(iii) softmax引起的条件密度中的紧密分子分母耦合。 我们的方法引入了与门控划分几何对齐的Voronoi型损失函数，并建立了最大似然估计量（MLE）的有限样本收敛速率。 在过参数化模型中，我们揭示了MLE的收敛速率与关联多项式方程组可解性的联系，这些方程组表征了接近不可识别的方向。 对于模型选择，我们将混合测度的树状图适应于SGMoE，得到一个一致的、无需扫描的专家数量选择器，在过拟合下达到点最优参数速率，同时避免多尺寸训练。 在合成数据上的模拟验证了理论，准确恢复了专家数量，并实现了参数估计的预测速率，同时密切逼近回归函数。 在模型误指的情况下（例如$\epsilon$污染），树状图选择准则具有鲁棒性，恢复了真实的混合成分数量，而Akaike信息准则、贝叶斯信息准则和综合完成似然则随着样本量增加倾向于过度选择。 在干旱响应性状的玉米蛋白质组学数据集上，我们的树状图引导的SGMoE选择了两个专家，揭示了清晰的混合测度层次结构，早期稳定了似然，并产生了可解释的基因型-表型图，优于标准准则，且无需多尺寸训练。 显示英文摘要

摘要： We develop a unified statistical framework for softmax-gated Gaussian mixture of experts (SGMoE) that addresses three long-standing obstacles in parameter estimation and model selection: (i) non-identifiability of gating parameters up to common translations, (ii) intrinsic gate-expert interactions that induce coupled differential relations in the likelihood, and (iii) the tight numerator-denominator coupling in the softmax-induced conditional density. Our approach introduces Voronoi-type loss functions aligned with the gate-partition geometry and establishes finite-sample convergence rates for the maximum likelihood estimator (MLE). In over-specified models, we reveal a link between the MLE's convergence rate and the solvability of an associated system of polynomial equations characterizing near-nonidentifiable directions. For model selection, we adapt dendrograms of mixing measures to SGMoE, yielding a consistent, sweep-free selector of the number of experts that attains pointwise-optimal parameter rates under overfitting while avoiding multi-size training. Simulations on synthetic data corroborate the theory, accurately recovering the expert count and achieving the predicted rates for parameter estimation while closely approximating the regression function. Under model misspecification (e.g., $\epsilon$-contamination), the dendrogram selection criterion is robust, recovering the true number of mixture components, while the Akaike information criterion, the Bayesian information criterion, and the integrated completed likelihood tend to overselect as sample size grows. On a maize proteomics dataset of drought-responsive traits, our dendrogram-guided SGMoE selects two experts, exposes a clear mixing-measure hierarchy, stabilizes the likelihood early, and yields interpretable genotype-phenotype maps, outperforming standard criteria without multi-size training.