Loop Corrections in Spin Models through Density Consistency
Abstract
Computing marginal distributions of discrete or semidiscrete Markov random fields (MRFs) is a fundamental, generally intractable problem with a vast number of applications in virtually all fields of science. We present a new family of computational schemes to approximately calculate the marginals of discrete MRFs. This method shares some desirable properties with belief propagation, in particular, providing exact marginals on acyclic graphs, but it differs with the latter in that it includes some loop corrections; i.e., it takes into account correlations coming from all cycles in the factor graph. It is also similar to the adaptive Thouless-Anderson-Palmer method, but it differs with the latter in that the consistency is not on the first two moments of the distribution but rather on the value of its density on a subset of values. The results on finite-dimensional Isinglike models show a significant improvement with respect to the Bethe-Peierls (tree) approximation in all cases and with respect to the plaquette cluster variational method approximation in many cases. In particular, for the critical inverse temperature βc of the homogeneous hypercubic lattice, the expansion of (dβc) -1 around d =∞ of the proposed scheme is exact up to d-4 order, whereas the latter two are exact only up to d-2 order.
- Publication:
-
Physical Review Letters
- Pub Date:
- July 2019
- DOI:
- 10.1103/PhysRevLett.123.020604
- arXiv:
- arXiv:1810.10602
- Bibcode:
- 2019PhRvL.123b0604B
- Keywords:
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- Condensed Matter - Statistical Mechanics;
- Condensed Matter - Disordered Systems and Neural Networks;
- Computer Science - Machine Learning;
- Physics - Computational Physics
- E-Print:
- 12 pages, 3 figures, 1 table