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Classical matrix perturbation results, such as Weyl's theorem for eigenvaluesand the Davis-Kahan theorem for eigenvectors, are general purpose. Theseclassical bounds are tight in the worst case, but in many settings sub-optimalin the typical case. In this paper, we present perturbation bounds whichconsider the nature of the perturbation and its interaction with theunperturbed structure in order to obtain significant improvements over theclassical theory in many scenarios, such as when the perturbation is random. Wedemonstrate the utility of these new results by analyzing perturbations in thestochastic blockmodel where we derive much tighter bounds than provided by theclassical theory. We use our new perturbation theory to show that a very simpleand natural clustering algorithm -- whose analysis was difficult using theclassical tools -- nevertheless recovers the communities of the blockmodelexactly even in very sparse graphs.

Unperturbed: spectral analysis beyond Davis-Kahan