Approximate inference on planar graphs using loop calculus and belief propagation

Vicenç Gómez, Hilbert J. Kappen, Michael Chertkov

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


We introduce novel results for approximate inference on planar graphical models using the loop calculus framework. The loop calculus (Chertkov and Chernyak, 2006a) allows to express the exact partition function of a graphical model as a finite sum of terms that can be evaluated once the belief propagation (BP) solution is known. In general, full summation over all correction terms is intractable. We develop an algorithm for the approach presented in Chertkov et al. (2008) which represents an efficient truncation scheme on planar graphs and a new representation of the series in terms of Pfaffians of matrices. We analyze the performance of the algorithm for models with binary variables and pairwise interactions on grids and other planar graphs. We study in detail both the loop series and the equivalent Pfaffian series and show that the first term of the Pfaffian series for the general, intractable planar model, can provide very accurate approximations. The algorithm outperforms previous truncation schemes of the loop series and is competitive with other state of the art methods for approximate inference.

Original languageEnglish (US)
Pages (from-to)1273-1296
Number of pages24
JournalJournal of Machine Learning Research
StatePublished - Apr 2010
Externally publishedYes


  • Approximate inference
  • Belief propagation
  • Ising model
  • Loop calculus
  • Partition function
  • Planar graphs

ASJC Scopus subject areas

  • Software
  • Artificial Intelligence
  • Control and Systems Engineering
  • Statistics and Probability


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