### 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 language | English (US) |
---|---|

Pages (from-to) | 1-35 |

Number of pages | 35 |

Journal | Foundations of Computational Mathematics |

DOIs | |

State | Accepted/In press - Nov 17 2016 |

### Fingerprint

### Keywords

- Convex relaxation
- Laplacian matrices
- Random matrices
- Semidefinite programming

### ASJC Scopus subject areas

- Analysis
- Computational Theory and Mathematics
- Computational Mathematics
- Applied Mathematics

### Cite this

*Foundations of Computational Mathematics*, 1-35. https://doi.org/10.1007/s10208-016-9341-9

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

Research output: Contribution to journal › Article

*Foundations of Computational Mathematics*, pp. 1-35. https://doi.org/10.1007/s10208-016-9341-9

}

TY - JOUR

T1 - Random Laplacian Matrices and Convex Relaxations

AU - Bandeira, Afonso

PY - 2016/11/17

Y1 - 2016/11/17

N2 - 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.

AB - 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.

KW - Convex relaxation

KW - Laplacian matrices

KW - Random matrices

KW - Semidefinite programming

UR - http://www.scopus.com/inward/record.url?scp=84995752841&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84995752841&partnerID=8YFLogxK

U2 - 10.1007/s10208-016-9341-9

DO - 10.1007/s10208-016-9341-9

M3 - Article

AN - SCOPUS:84995752841

SP - 1

EP - 35

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

ER -