J M Mooij and H J Kappen
Department of Biophysics, Institute for Neuroscience, Radboud University Nijmegen, Geert Grooteplein 21, 6525 EZ Nijmegen, The Netherlands
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J M Mooij and H J Kappen J. Stat. Mech. (2005) P11012
Part of Focus on Optimization and Inference in Machine Learning and Physics
We analyse the local stability of the high-temperature fixed point of the loopy belief propagation (LBP) algorithm and how this relates to the properties of the Bethe free energy which LBP tries to minimize. We focus on the case of binary networks with pairwise interactions. In particular, we state sufficient conditions for convergence of LBP to a unique fixed point and show that these are sharp for purely ferromagnetic interactions. In contrast, in the purely antiferromagnetic case, the undamped parallel LBP algorithm is suboptimal in the sense that the stability of the fixed point breaks down much earlier than for damped or sequential LBP; we observe that the onset of instability for the latter algorithms is related to the properties of the Bethe free energy. For spin-glass interactions, damping LBP only helps slightly. We estimate analytically the temperature at which the high-temperature LBP fixed point becomes unstable for random graphs with arbitrary degree distributions and random interactions.
75.10.Nr Spin-glass and other random models
75.30.Kz Magnetic phase boundaries (including magnetic transitions, metamagnetism, etc.)
02.10.Ox Combinatorics; graph theory
65.60.+a Thermal properties of amorphous solids and glasses: heat capacity, thermal expansion, etc.
82D30 Random media, disordered materials (including liquid crystals and spin glasses)
Issue 11 (November 2005)
Received 19 May 2005 , accepted for publication 6 October 2005
Published 30 November 2005
J M Mooij and H J Kappen J. Stat. Mech. (2005) P11012
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