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Modern applications in statistics, computer science and network science haveseen tremendous values of finer matrix spectral perturbation theory. In thispaper, we derive a generic $\ell_{2\rightarrow\infty}$ eigenspace perturbationbound for symmetric random matrices, with independent or dependent entries andfairly flexible entry distributions. In particular, we apply our generic boundto binary random matrices with independent entries or with certain dependencystructures, including the unnormalized Laplacian of inhomogenous random graphsand $m$-dependent matrices. Through a detailed comparison, we found that forbinary random matrices with independent entries, our$\ell_{2\rightarrow\infty}$ bound is tighter than all existing bounds that weare aware of, while our condition is weaker than all but one of them in a lesscommon regime. We apply our perturbation bounds in three problems and improvethe state of the art: concentration of the spectral norm of sparse randomgraphs, exact recovery of communities in stochastic block models and partialconsistency of divisive hierarchical clustering. Finally we discuss theextensions of our theory to random matrices with more complex dependencystructures and non-binary entries, asymmetric rectangular matrices and inducedperturbation theory in other metrics.

Unified $\ell_{2\rightarrow\infty}$ Eigenspace Perturbation Theory for Symmetric Random Matrices