The singular subspaces perturbation theory is of fundamental importance in probability and statistics. It has various applications across different fields. We consider two arbitrary matrices where one is a leave-one-column-out submatrix of the other one and establish a novel perturbation upper bound for the distance between the two corresponding singular subspaces. It is well-suited for mixture models and results in a sharper and finer statistical analysis than classical perturbation bounds such as Wedin's Theorem. Empowered by this leave-one-out perturbation theory, we provide a deterministic entrywise analysis for the performance of spectral clustering under mixture models. Our analysis leads to an explicit exponential error rate for spectral clustering of sub-Gaussian mixture models. For the mixture of isotropic Gaussians, the rate is optimal under a weaker signal-to-noise condition than that of L{\"o}ffler et al. (2021).

翻译：奇异子空间扰动理论在概率与统计中具有基础性重要性，并在不同领域具有广泛应用。我们考虑两个任意矩阵，其中一个是另一个的留一列子矩阵，并建立两个对应奇异子空间之间距离的新扰动上界。该方法特别适用于混合模型，并比经典扰动界（如韦丁定理）产生更尖锐、更精细的统计分析。基于这一留一法扰动理论，我们提供了混合模型下谱聚类性能的确定性逐元素分析。我们的分析得出了亚高斯混合模型谱聚类的显式指数误差率。对于各向同性高斯混合，该速率在比Löffler等人（2021）更弱的信噪比条件下达到最优。