Optimal estimation of high-dimensional Gaussian location mixtures
Optimal estimation of high-dimensional Gaussian location mixtures
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DOI：
10.1214/22-aos2207
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发表时间：
2020-02
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影响因子：
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通讯作者：
Natalie Doss;Yihong Wu;Pengkun Yang;Harrison H. Zhou
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文献类型：
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作者：
Natalie Doss;Yihong Wu;Pengkun Yang;Harrison H. Zhou
本文研究了高维有限高斯位置混合模型在无分离条件下的最优估计率。我们假设组分的数量$k$是有界的，中心位于有界半径的球中，同时允许维度$d$与样本大小$n$一样大。推广了Heinrich和Kahn \cite{HK 2015}的一维结果，我们证明了估计Wasserstein距离的混合分布的极大极小化率为$\Theta（（d/n）^{1/4} + n^{-1/（4k-2）}）$，可通过时间$O（nd^2+n^{5/4}）$计算得到。此外，我们表明，混合密度可以估计在最佳参数率$\Theta（\sqrt{d/n}）$在Hellinger距离，并提供了一个计算效率高的算法，以实现这一速度在特殊情况下的k=2$。理论和方法的发展都依赖于矩量法的谨慎应用。我们的研究结果的核心是观察到有限高斯混合物的信息几何的特征在于混合分布的矩张量，其低秩结构可以被利用来获得一个尖锐的局部熵界。
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 $\Theta((d/n)^{1/4} + n^{-1/(4k-2)})$, 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 $\Theta(\sqrt{d/n})$ 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.