We study the relation between the total variation (TV) and Hellinger distances between two Gaussian location mixtures. Our first result establishes a general upper bound: for any two mixing distributions supported on a compact set, the Hellinger distance between the two mixtures is controlled by the TV distance raised to a power $1-o(1)$, where the $o(1)$ term is of order $1/\log\log(1/\mathrm{TV})$. We also construct two sequences of mixing distributions that demonstrate the sharpness of this bound. Taken together, our results resolve an open problem raised in Jia et al. (2023) and thus lead to an entropic characterization of learning Gaussian mixtures in total variation. Our inequality also yields optimal robust estimation of Gaussian mixtures in Hellinger distance, which has a direct implication for bounding the minimax regret of empirical Bayes under Huber contamination.

翻译：我们研究两个高斯位置混合模型之间的总变差（TV）距离与海林格距离的关系。我们的第一个结果建立了一个普遍上界：对于支撑在紧集上的任意两个混合分布，两个混合模型间的海林格距离受总变差距离的$1-o(1)$次幂控制，其中$o(1)$项的量级为$1/\log\log(1/\mathrm{TV})$。我们还构造了两个混合分布序列以证明该界限的尖锐性。综合来看，我们的结果解决了Jia等人（2023）提出的一个开放问题，从而为总变差意义下高斯混合模型的学习提供了熵特征刻画。我们的不等式还导出了海林格距离下高斯混合模型的最优鲁棒估计，这对Huber污染下经验贝叶斯极小极大遗憾的界具有直接意义。