The method of moments is a statistical technique for density estimation that solves a system of moment equations to estimate the parameters of an unknown distribution. A fundamental question critical to understanding identifiability asks how many moment equations are needed to get finitely many solutions and how many solutions there are. We answer this question for classes of Gaussian mixture models using the tools of polyhedral geometry. Using these results, we present an algorithm that performs parameter recovery, and therefore density estimation, for high dimensional Gaussian mixture models that scales linearly in the dimension.

翻译：矩方法是密度估计中的一种统计技术，通过求解矩方程组来估计未知分布的参数。一个对理解可辨识性至关重要的基本问题涉及：需要多少个矩方程才能得到有限多个解，以及这些解的数量是多少。我们利用多面体几何工具，对高斯混合模型类别回答了这一问题。基于这些结果，我们提出了一种算法，该算法能够对高维高斯混合模型进行参数恢复，从而实现密度估计，且其复杂度随维度线性增长。