Random Laplacian Matrices and Convex Relaxations

• Published in: Foundations of computational mathematics Vol. 18; no. 2; pp. 345 - 379
• Main Author: Bandeira, Afonso S.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2018; Springer
• Subjects: Algorithms; Applications of Mathematics; Computer Science; Economics; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical research; Mathematics; Mathematics and Statistics; Matrices (Mathematics); Matrix Theory; Numerical Analysis; Regression analysis; 15B52; Random matrices; Semidefinite programming; 05C85; Convex relaxation; 05C80; Laplacian matrices
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-016-9341-9

The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a class of random Laplacian matrices with independent off-diagonal entries, this bound is essentially tight: the largest eigenvalue is, up to lower order terms, often the size of the largest diagonal. entry. Besides being a simple tool to obtain precise estimates on the largest eigenvalue of a class of random Laplacian matrices, our main result settles a number of open problems related to the tightness of certain convex relaxation-based algorithms. It easily implies the optimality of the semidefinite relaxation approaches to problems such as Z 2 Synchronization and stochastic block model recovery. Interestingly, this result readily implies the connectivity threshold for Erdős–Rényi graphs and suggests that these three phenomena are manifestations of the same underlying principle. The main tool is a recent estimate on the spectral norm of matrices with independent entries by van Handel and the author.