TAP equation for non-negative Boltzmann machine

Muneki Yasuda, Kazuyuki Tanaka

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


Mean-field methods for spin systems are frequently used in not only statistical physics but also information sciences. We focus on the Plefka expansion method for spin systems with two-body interactions. The Plefka expansion is a useful perturbative expansion of the Gibbs free energy, and it can systematically provide the naive mean-field approximation, the Thouless-Anderson-Palmer (TAP) equation and higher-order approximations. In the first part of this paper, using the linear response relation, we derive a recurrence formula for perturbative coefficients in the Plefka expansion. Our recurrence formula enables us to systematically derive general order coefficients. In the latter part of the paper, we apply our recurrence formula to the non-negative Boltzmann machine in which all spin variables are constrained to have non-negative real values, and we obtain the TAP equation for this model. We verify the performance of our TAP equation by conducting some numerical experiments.

Original languageEnglish
Pages (from-to)192-209
Number of pages18
JournalPhilosophical Magazine
Issue number1-3
Publication statusPublished - 2012 Jan 1


  • linear response relation
  • Markov random fields
  • mean-field methods
  • non-negative Boltzmann machines
  • Plefka expansion
  • TAP equation


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