Leave-one-out Singular Subspace Perturbation Analysis for Spectral Clustering

• URL: http://arxiv.org/abs/2205.14855v2
• Date: Sun, 14 Jan 2024 06:07:45 GMT
• Title: Leave-one-out Singular Subspace Perturbation Analysis for Spectral Clustering
• Authors: Anderson Y. Zhang, Harrison H. Zhou
• Abstract summary: The singular subspaces perturbation theory is of fundamental importance in probability and statistics. We consider two arbitrary matrices where one is a leave-one-column-out submatrix of the other one. 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.
• Score: 7.342677574855651
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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).

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