Optimal estimation of high-dimensional location Gaussian mixtures
This provides foundational statistical theory for mixture models in high dimensions, with potential applications in fields like machine learning and data science, though it is incremental as it extends prior one-dimensional results.
The paper tackles the problem of estimating high-dimensional Gaussian location mixtures without separation conditions, showing that the minimax rate for the mixing distribution in Wasserstein distance is Θ((d/n)^{1/4} + n^{-1/(4k-2)}) and that the mixture density can be estimated at the optimal parametric rate Θ(√(d/n)) in Hellinger distance.
This paper studies the optimal rate of estimation in a finite Gaussian location mixture model in high dimensions without separation conditions. We assume that the number of components $k$ is bounded and that the centers lie in a ball of bounded radius, while allowing the dimension $d$ to be as large as the sample size $n$. Extending the one-dimensional result of Heinrich and Kahn \cite{HK2015}, we show that the minimax rate of estimating the mixing distribution in Wasserstein distance is $Θ((d/n)^{1/4} + n^{-1/(4k-2)})$, achieved by an estimator computable in time $O(nd^2+n^{5/4})$. Furthermore, we show that the mixture density can be estimated at the optimal parametric rate $Θ(\sqrt{d/n})$ in Hellinger distance and provide a computationally efficient algorithm to achieve this rate in the special case of $k=2$. Both the theoretical and methodological development rely on a careful application of the method of moments. Central to our results is the observation that the information geometry of finite Gaussian mixtures is characterized by the moment tensors of the mixing distribution, whose low-rank structure can be exploited to obtain a sharp local entropy bound.