Statistics Papers

Document Type

Journal Article

Date of this Version

11-2009

Publication Source

Combinatorics, Probability and Computing

Volume

18

Issue

6

Start Page

881

Last Page

912

DOI

10.1017/S096354830900981X

Abstract

Belief propagation (BP) is a message-passing algorithm that computes the exact marginal distributions at every vertex of a graphical model without cycles. While BP is designed to work correctly on trees, it is routinely applied to general graphical models that may contain cycles, in which case neither convergence, nor correctness in the case of convergence is guaranteed. Nonetheless, BP has gained popularity as it seems to remain effective in many cases of interest, even when the underlying graph is ‘far’ from being a tree. However, the theoretical understanding of BP (and its new relative survey propagation) when applied to CSPs is poor.

Contributing to the rigorous understanding of BP, in this paper we relate the convergence of BP to spectral properties of the graph. This encompasses a result for random graphs with a ‘planted’ solution; thus, we obtain the first rigorous result on BP for graph colouring in the case of a complex graphical structure (as opposed to trees). In particular, the analysis shows how belief propagation breaks the symmetry between the 3! possible permutations of the colour classes.

Keywords

belief Propagation, survey propagation, graph coloring, spectral algorithms.

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Date Posted: 27 November 2017

This document has been peer reviewed.