Random Laplacian Matrices and Convex Relaxations

Afonso 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 (Formula presented.) 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)1-35
Number of pages35
JournalFoundations of Computational Mathematics
DOIs
StateAccepted/In press - Nov 17 2016

Fingerprint

Convex Relaxation
Laplacian Matrix
Largest Eigenvalue
Random Matrices
Semidefinite Relaxation
Spectral Norm
Imply
Tightness
Estimate
Open Problems
Optimality
Connectivity
Synchronization
Recovery
Term
Graph in graph theory
Class
Model

Keywords

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

ASJC Scopus subject areas

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

Cite this

Random Laplacian Matrices and Convex Relaxations. / Bandeira, Afonso.

In: Foundations of Computational Mathematics, 17.11.2016, p. 1-35.

Research output: Contribution to journalArticle

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