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This paper studies the optimal rate of estimation in a finite Gaussianlocation mixture model in high dimensions without separation conditions. Weassume that the number of components $k$ is bounded and that the centers lie ina ball of bounded radius, while allowing the dimension $d$ to be as large asthe sample size $n$. Extending the one-dimensional result of Heinrich and Kahn\cite{HK2015}, we show that the minimax rate of estimating the mixingdistribution in Wasserstein distance is $\Theta((d/n)^{1/4} + n^{-1/(4k-2)})$,achieved by an estimator computable in time $O(nd^2+n^{5/4})$. Furthermore, weshow that the mixture density can be estimated at the optimal parametric rate$\Theta(\sqrt{d/n})$ in Hellinger distance and provide a computationallyefficient algorithm to achieve this rate in the special case of $k=2$.  Both the theoretical and methodological development rely on a carefulapplication of the method of moments. Central to our results is the observationthat the information geometry of finite Gaussian mixtures is characterized bythe moment tensors of the mixing distribution, whose low-rank structure can beexploited to obtain a sharp local entropy bound.

Optimal estimation of high-dimensional location Gaussian mixtures