Spectral methods have been widely used for a large class of challenging problems, ranging from top-K ranking via pairwise comparisons, community detection, factor analysis, among others.

Analyses of these spectral methods require super-norm perturbation analysis of top eigenvectors. This allows us to UNIFORMLY approximate elements in eigenvectors by linear functions of the observed random matrix that can be analyzed further. We first establish such an infinity-norm pertubation bound for top eigenvectors and apply the idea to several challenging problems such as top-$K$ ranking, community detections, $Z_2$-syncronization and matrix completion. We show that the spectral methods are indeed optimal for these problems. We illustrate these methods via simulations.

(Based on joint work with Emmanuel Abbe, Kaizheng Wang, Yiqiao Zhong and that of Yixin Chen, Cong Ma and Kaizheng Wang)