Afonso S. Bandeira

Exact Recovery in the Stochastic Block Model

The stochastic block model with two communities, or equivalently the planted bisection model, is a popular model of random graph exhibiting a cluster behavior. In the symmetric case, the graph has two equally sized clusters and vertices connect with probability p within clusters and q across clusters. In the past two decades, a large body of literature in statistics and computer science has focused on providing lower bounds on the scaling of | p - q| to ensure exact recovery. In this paper, we identify a sharp threshold phenomenon for exact recovery: if α = pn/log(n) and β = qn/ log(n) are constant (with α > β), recovering the communities with high probability is possible if (α + β/2) - √(αβ) > 1 and is impossible if (α + β/2) - √(αβ) <; 1. In particular, this improves the existing bounds. This also sets a new line of sight for efficient clustering algorithms. While maximum likelihood (ML) achieves the optimal threshold (by definition), it is in the worst case NP-hard. This paper proposes an efficient algorithm based on a semidefinite programming relaxation of ML, which is proved to succeed in recovering the communities close to the threshold, while numerical experiments suggest that it may achieve the threshold. An efficient algorithm that succeeds all the way down to the threshold is also obtained using a partial recovery algorithm combined with a local improvement procedure.

Phase Retrieval with Polarization

In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and exploits certain properties of expander graphs. We also give an efficient phase retrieval procedure, and use recent results in spectral graph theory to produce a stable performance guarantee which rivals the guarantee for PhaseLift in [Candes et al. 2011]. We use numerical simulations to illustrate the performance of our phase retrieval procedure, and we compare reconstruction error and runtime with a common alternating-projections-type procedure.

Certifying the Restricted Isometry Property is Hard

This paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property (RIP). We demonstrate that testing whether a matrix satisfies RIP is NP-hard. As a consequence of our result, it is impossible to efficiently test for RIP provided P ≠ NP.

A Cheeger Inequality for the Graph Connection Laplacian

The

Sharp nonasymptotic bounds on the norm of random matrices with independent entries

O(d)

Decoding Binary Node Labels from Censored Edge Measurements: Phase Transition and Efficient Recovery

synchronization problem consists of estimating a set of

Tightness of the maximum likelihood semidefinite relaxation for angular synchronization

n

Relax, No Need to Round: Integrality of Clustering Formulations

unknown orthogonal

Random Laplacian Matrices and Convex Relaxations

d\times d

Multireference alignment using semidefinite programming

matrices