Random Laplacian matrices and convex relaxations
Abstract
The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a large class of random Laplacian matrices, 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 large 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 $\mathbb{Z}_2$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2015
- DOI:
- 10.48550/arXiv.1504.03987
- arXiv:
- arXiv:1504.03987
- Bibcode:
- 2015arXiv150403987B
- Keywords:
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- Mathematics - Probability;
- Computer Science - Data Structures and Algorithms;
- Computer Science - Social and Information Networks;
- Mathematics - Optimization and Control