We quantify the parameter stability of a spherical Gaussian Mixture Model (sGMM) under small perturbations in distribution space. Namely, we derive the first explicit bound to show that for a mixture of spherical Gaussian $P$ (sGMM) in a pre-defined model class, all other sGMM close to $P$ in this model class in total variation distance has a small parameter distance to $P$. Further, this upper bound only depends on $P$. The motivation for this work lies in providing guarantees for fitting Gaussian mixtures; with this aim in mind, all the constants involved are well defined and distribution free conditions for fitting mixtures of spherical Gaussians. Our results tighten considerably the existing computable bounds, and asymptotically match the known sharp thresholds for this problem.

翻译：我们量化了分布空间中小扰动下球面高斯混合模型(sGMM)的参数稳定性。具体而言，我们推导出首个显式界，表明对于预定义模型类中的球面高斯混合模型$P$（sGMM），该模型类中所有与$P$在总变差距离上接近的其他sGMM，其与$P$的参数距离都很小。进一步地，该上界仅依赖于$P$。本研究动机在于为高斯混合拟合提供保障；基于此目标，所有涉及常数均定义良好，且拟合球面高斯混合的条件与分布无关。我们的结果显著紧化了现有可计算界，并在渐近意义上与该问题的已知尖锐阈值相匹配。