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Broadcasting with Random Matrices

Authors Charilaos Efthymiou, Kostas Zampetakis

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Author Details

Charilaos Efthymiou
  • Computer Science, University of Warwick, Coventry, UK
Kostas Zampetakis
  • Computer Science, University of Warwick, Coventry, UK

Funding

  • Efthymiou, Charilaos: EPSRC New Investigator Award (grant no. EP/V050842/1) and Centre of Discrete Mathematics and Applications (DIMAP), The University of Warwick.
  • Zampetakis, Kostas: EPSRC New Investigator Award (grant no. EP/V050842/1) and Centre of Discrete Mathematics and Applications (DIMAP), The University of Warwick.

Acknowledgements

We are grateful to the anonymous reviewers for their thorough review of our submission, and for their insightful comments and suggestions.

Cite As Get BibTex

Charilaos Efthymiou and Kostas Zampetakis. Broadcasting with Random Matrices. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 55:1-55:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.55

Abstract

Motivated by the theory of spin-glasses in physics, we study the so-called reconstruction problem on the tree, and on the sparse random graph G(n,d/n). Both cases reduce naturally to analysing broadcasting models, where each edge has its own broadcasting matrix, and this matrix is drawn independently from a predefined distribution. 
We establish the reconstruction threshold for the cases where the broadcasting matrices give rise to symmetric, 2-spin Gibbs distributions. This threshold seems to be a natural extension of the well-known Kesten-Stigum bound that manifests in the classic version of the reconstruction problem. Our results determine, as a special case, the reconstruction threshold for the prominent Edwards–Anderson model of spin-glasses, on the tree. 
Also, we extend our analysis to the setting of the Galton-Watson random tree, and the (sparse) random graph G(n,d/n), where we establish the corresponding thresholds. Interestingly, for the Edwards–Anderson model on the random graph, we show that the replica symmetry breaking phase transition, established by Guerra and and Toninelli in [Guerra and Toninelli, 2004], coincides with the reconstruction threshold. 
Compared to classical Gibbs distributions, spin-glasses have several unique features. In that respect, their study calls for new ideas, e.g. we introduce novel estimators for the reconstruction problem. The main technical challenge in the analysis of such systems, is the presence of (too) many levels of randomness, which we manage to circumvent by utilising recently proposed tools coming from the analysis of Markov chains.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Discrete mathematics
Keywords
  • spin-system
  • spin-glass
  • sparse random graph
  • reconstruction
  • phase transitions

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