  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).
