Leave-one-out Singular Subspace Perturbation Analysis for Spectral Clustering
arxiv(2022)
摘要
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).
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关键词
singular subspace perturbation analysis,spectral
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