Random Laplacian Matrices and Convex Relaxations

Afonso S. Bandeira

Research output: Contribution to journalArticle

Abstract

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 Z2 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.

Original languageEnglish (US)
Pages (from-to)345-379
Number of pages35
JournalFoundations of Computational Mathematics
Volume18
Issue number2
DOIs
StatePublished - Apr 1 2018

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Keywords

  • Convex relaxation
  • Laplacian matrices
  • Random matrices
  • Semidefinite programming

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics

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