We study an open problem of understanding the effects of the minimum component separation on the convergence rates of parameter estimation in finite Gaussian mixtures. We address this by developing a unified geometric framework based on novel Hellinger lower bounds that directly relate discrepancies between mixture densities directly to Wasserstein distances between their underlying mixing measures, with explicit dependence on both the minimum separation and the minimum weight. Our approach combines carefully designed interpolation polynomials with confluent divided difference techniques to construct specialized moment-extraction test functions. When the number of components is known, these bounds uncover a localization phenomenon: the separation complexity is driven strictly by the spatial configuration of mixture components, namely, whether they are concentrated in a single cluster, partitioned into multiple clusters separated by a macroscopic gap, or arranged without any structural constraints. On the other hand, when the number of components becomes unknown and is over-specified, the separation complexity is slightly reduced, while the minimum mixture weight disappears entirely from the convergence rates due to a transition from first-order to second-order Wasserstein geometry. As a consequence, we obtain separation-dependent convergence rates that continuously interpolate between point-wise and uniform estimation regimes, thereby settling the fundamental limits of parameter recovery in finite Gaussian mixtures.

翻译：我们研究了一个开放性问题：理解最小成分分离对有限高斯混合中参数估计收敛速度的影响。为此，我们发展了一个基于新颖Hellinger下界的统一几何框架，该框架直接将混合密度之间的差异与潜在混合测度之间的Wasserstein距离联系起来，并显式依赖于最小分离和最小权重。我们的方法结合了精心设计的插值多项式与汇合差商技术，构造了专用的矩提取测试函数。当成分数量已知时，这些界揭示了一种局域化现象：分离复杂度完全由混合成分的空间构型决定，即它们是否集中在一个簇中、被宏观间隙分隔成多个簇、或排列无任何结构约束。另一方面，当成分数量未知且被过度指定时，分离复杂度略有降低，而最小混合权重因从一阶Wasserstein几何向二阶Wasserstein几何的转变而从收敛速度中完全消失。由此，我们得到了依赖分离度的收敛速度，该速度在逐点估计与一致估计区间之间连续插值，从而确立了有限高斯混合中参数恢复的基本极限。