Leave-one-out singular subspace perturbation analysis for spectral clustering
研究了留一列子矩阵的奇异子空间扰动上界，比经典Wedin定理更精细，用于谱聚类在混合模型下的确定性逐元素分析，得到次高斯混合模型谱聚类的显式指数误差率。
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 Löffler et al. (2021).