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., $ε$-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.

翻译：我们为Softmax门控高斯混合专家模型（SGMoE）建立了一个统一的统计框架，解决了参数估计与模型选择中三个长期存在的障碍：（i）门控参数在公共平移下的不可辨识性；（ii）内在的门控-专家相互作用导致似然函数中的耦合微分关系；以及（iii）Softmax诱导的条件密度中分子与分母的紧耦合。我们引入了与门控分区几何结构对齐的Voronoi型损失函数，并建立了极大似然估计（MLE）的有限样本收敛速率。在过指定模型中，我们揭示了MLE收敛速率与表征近不可辨识方向的多项式方程组可解性之间的关联。针对模型选择，我们将混合度量树状图方法适配至SGMoE，得到了一致且无需扫描的专家数量选择器：该选择器在避免多尺度训练的同时，于过拟合情形下实现了参数率的逐点最优。合成数据上的仿真验证了理论分析：准确恢复专家数量并达成参数估计的预测速率，同时紧密逼近回归函数。在模型误设定（如ε-污染）条件下，树状图选择准则保持稳健，能够恢复真实的混合成分数量，而赤池信息准则、贝叶斯信息准则和集成完全似然准则随着样本量增大会倾向于过度选择。在一个关于干旱响应性状的玉米蛋白质组学数据集中，我们基于树状图引导的SGMoE选择了两个专家，揭示了清晰的混合度量层级结构，使似然函数提前稳定，并生成了可解释的基因型-表型图谱，其性能优于无需多尺度训练的标准准则。