Linear inverse problems on Erdos-Rényi graphs: Information-theoretic limits and efficient recovery

Emmanuel Abbe, Afonso S. Bandeira, Annina Bracher, Amit Singer

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

This paper considers the inverse problem with observed variables Y = B GX +Z, where BG is the incidence matrix of a graph G, X is the vector of unknown vertex variables with a uniform prior, and Z is a noise vector with Bernoulli(ε) i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery of X is possible if and only the graph G is connected, with a sharp threshold at the edge probability log(n)=n for Erdos-Rényi random graphs. The first goal of this paper is to determine how the edge probability p needs to scale to allow exact recovery in the presence of noise. Defining the degree (oversampling) rate of the graph by α = np= log(n), it is shown that exact recovery is possible if and only if α > 2/(1-2ε)2+o(1/(1-2ε)2). In other words, 2/(1-2ε)2 is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. Full version available in [1].

Original languageEnglish (US)
Title of host publication2014 IEEE International Symposium on Information Theory, ISIT 2014
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages1251-1255
Number of pages5
ISBN (Print)9781479951864
DOIs
StatePublished - Jan 1 2014
Event2014 IEEE International Symposium on Information Theory, ISIT 2014 - Honolulu, HI, United States
Duration: Jun 29 2014Jul 4 2014

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
ISSN (Print)2157-8095

Other

Other2014 IEEE International Symposium on Information Theory, ISIT 2014
CountryUnited States
CityHonolulu, HI
Period6/29/147/4/14

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Information Systems
  • Modeling and Simulation
  • Applied Mathematics

Cite this

Abbe, E., Bandeira, A. S., Bracher, A., & Singer, A. (2014). Linear inverse problems on Erdos-Rényi graphs: Information-theoretic limits and efficient recovery. In 2014 IEEE International Symposium on Information Theory, ISIT 2014 (pp. 1251-1255). [6875033] (IEEE International Symposium on Information Theory - Proceedings). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ISIT.2014.6875033