We investigate the landscape of the negative log-likelihood function of Gaussian Mixture Models (GMMs) with a general number of components in the population limit. As the objective function is non-convex, there can be multiple local minima that are not globally optimal, even for well-separated mixture models. Our study reveals that all local minima share a common structure that partially identifies the cluster centers (i.e., means of the Gaussian components) of the true location mixture. Specifically, each local minimum can be represented as a non-overlapping combination of two types of sub-configurations: fitting a single mean estimate to multiple Gaussian components or fitting multiple estimates to a single true component. These results apply to settings where the true mixture components satisfy a certain separation condition, and are valid even when the number of components is over- or under-specified. We also present a more fine-grained analysis for the setting of one-dimensional GMMs with three components, which provide sharper approximation error bounds with improved dependence on the separation.

翻译：本文研究了在总体极限下，具有一般分量数量的高斯混合模型（GMMs）负对数似然函数的景观。由于目标函数是非凸的，即使对于分离良好的混合模型，也可能存在多个并非全局最优的局部极小值。我们的研究表明，所有局部极小值都共享一个共同结构，该结构部分地识别了真实位置混合的聚类中心（即高斯分量的均值）。具体而言，每个局部极小值可表示为两种子配置的非重叠组合：将单个均值估计拟合到多个高斯分量，或将多个估计拟合到单个真实分量。这些结果适用于真实混合分量满足特定分离条件的情形，且即使在分量数量被高估或低估时仍然成立。此外，我们针对具有三个分量的二维高斯混合模型给出了更精细的分析，该分析提供了更尖锐的近似误差界，并改进了对分离度的依赖关系。