Unperturbed: spectral analysis beyond Davis-Kahan

Classical matrix perturbation results, such as Weyl’s theorem for eigenvalues andthe Davis-Kahan theorem for eigenvectors, are general purpose. These classicalbounds are tight in the worst case, but in many settings sub-optimal in the typicalcase. In this paper, we present perturbation bounds which consider the nature ofthe perturbation and its interaction with the unperturbed structure in order to ob-tain significant improvements over the classical theory in many scenarios, such aswhen the perturbation is random. We demonstrate the utility of these new resultsby analyzing perturbations in the stochastic blockmodel where we derive muchtighter bounds than provided by the classical theory. We use our new perturbationtheory to show that a very simple and natural clustering algorithm – whose analy-sis was difficult using the classical tools – nevertheless recovers the communitiesof the blockmodel exactly even in very sparse graphs.