<div>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.&#160; This allows us to UNIFORMLY approximate elements in eigenvectors&#160; by linear functions of the observed random matrix that can be analyzed further.&#160; 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.&#160; We show that the spectral methods are indeed optimal for these problems.&#160; We illustrate these methods via simulations.

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