Region Graph Partition Function Expansion and Approximate Free Energy Landscapes: Theory and Some Numerical Results

Journal of Statistical Physics

Volumn 148, Issue 3, 2012, Pages 513-547

  Zhou, Haijun ^a   Wang, Chuang ^a  

^a INSTITUTE OF THEORETICAL PHYSICS   (China)

Author keywords

Belief propagation; Bethe Peierls free energy; Edwards Anderson spin glass; Graphical model; Partition function expansion; Region graph

Indexed keywords

EID: 84865423021     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-012-0555-1     Document Type: Article

References (56)

1. An, G.: A note on the cluster variation method. J. Stat. Phys. 52, 727-734 (1988).
2. Anderson, P. W.: Spin glass vii: Spin glass as paradigm. Phys. Today March, 9-11 (1990).
3. Aurell, E., Ollion, C., Roudi, Y.: Dynamics and performance of susceptibility propagation on synthetic data. Eur. Phys. J. B 77, 587-595 (2010).
4. Bethe, H. A.: Statistical theory of superlattices. Proc. R. Soc. Lond. Ser. A 150, 552-575 (1935).
5. Braunstein, A., Zecchina, R.: Learning by message passing in networks of discrete synapses. Phys. Rev. Lett. 96, 030201 (2006).
6. Chertkov, M., Chernyak, V. Y.: Loop series for discrete statistical models on graphs. J. Stat. Mech. Theory Exp. P06009 (2006).
7. Cover, T. M., Thomas, J. A.: Elements of Information Theory. Wiley, New York (1991).
8. Domínguez, E., Lage-Catellanos, A., Mulet, R., Ricci-Tersenghi, F., Rizzo, T.: Characterizing and improving generalized belief propagation algorithms on the 2d Edwards-Anderson model. J. Stat. Mech. Theory Exp. P12007 (2011).
9. Donoho, D. L., Maleki, A., Montanari, A.: Message-passing algorithms for compressed sensing. Proc. Natl. Acad. Sci. USA 106, 18914-18919 (2009).
10. Ediger, M. D.: Spatially heterogeneous dynamics in supercooled liquids. Annu. Rev. Phys. Chem. 51, 99-128 (2000).
11. Edwards, S. F., Anderson, P. W.: Theory of spin glasses. J. Phys. F, Met. Phys. 5, 965-974 (1975).
12. Glotzer, S. C.: Spatially heterogeneous dynamics in liquids: insights from simulation. J. Non-Cryst. Solids 274, 342-355 (2000).
13. Hartmann, A. K., Weigt, W.: Phase Transitions in Combinatorial Optimization Problems. Wiley-VCH, Weinheim (2005).
14. Kabashima, Y., Saad, D.: Statistical mechanics of error-correcting codes. Europhys. Lett. 45, 97-103 (1999).
15. Kikuchi, R.: A theory of cooperative phenomena. Phys. Rev. 81, 988-1003 (1951).
16. Kramers, H. A., Wannier, G. H.: Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev. 60, 252-262 (1941).
17. Krzakala, F., Mézard, M., Sausset, F., Sun, Y. F., Zdeborová, L.: Statistical physics-based reconstruction in compressed sensing. arXiv: 1109. 4424 (2011).
18. Krzakala, F., Montanari, A., Ricci-Tersenghi, F., Semerjian, G., Zdeborova, L.: Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. Natl. Acad. Sci. USA 104, 10318-10323 (2007).
19. Kschischang, F. R., Frey, B. J., Loeliger, H. A.: Factor graphs and the sum-product algorithm. IEEE Trans. Inf. Theory 47, 498-519 (2001).
20. Lage-Castellanos, A., Mulet, R., Ricci-Tersenghi, F., Rizzo, T.: Inference algorithm for finite-dimensional spin glasses: Belief propagation on the dual lattice. Phys. Rev. E 84, 046706 (2011).
21. Lage-Castellanos, A., Mulet, R., Ricci-Tersenghi, F., Rizzo, T.: Replica cluster variational method: the replica symmetric solution for the 2d random bond Ising model. arXiv: 1204. 0439 (2012).
22. Mézard, M., Montanari, A.: Reconstruction on trees and spin glass transition. J. Stat. Phys. 124, 1317-1350 (2006).
23. Mézard, M., Montanari, A.: Information, Physics, and Computation. Oxford University Press, New York (2009).
24. Mézard, M., Mora, T.: Constraint satisfaction problems and neural networks: a statistical physics perspective. J. Physiol. Paris 103, 107-113 (2009).
25. Mézard, M., Parisi, G.: The Bethe lattice spin glass revisited. Eur. Phys. J. B 20, 217-233 (2001).
26. Mézard, M., Parisi, G., Virasoro, M. A.: SK model: the replica solution without replicas. Europhys. Lett. 1, 77-82 (1986).
27. Mézard, M., Parisi, G., Virasoro, M. A.: Spin Glass Theory and Beyond. World Scientific, Singapore (1987).
28. Mézard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297, 812-815 (2002).
29. Monasson, R.: Structural glass transition and the entropy of the metastable states. Phys. Rev. Lett. 75, 2847-2850 (1995).
30. Monasson, R.: Optimization problems and replica symmetry breaking in finite connectivity spin glasses. J. Phys. A, Math. Gen. 31, 513-529 (1998).
31. Montanari, A., Rizzo, T.: How to compute loop corrections to Bethe approximation. J. Stat. Mech. Theory Exp. P10011 (2005).
32. Montanari, A., Semerjian, G.: On the dynamics of the glass transition on Bethe lattices. J. Stat. Phys. 124, 103-189 (2006).
33. Morgenstern, I., Binder, K.: Magnetic correlations in two-dimensional spin-glasses. Phys. Rev. B 22, 288-303 (1980).
34. Morita, T., Suzuki, M., Wada, K., Kaburagi, M. (eds.): Foundations and applications of cluster variation method and path probability method. Prog. Theor. Phys., Suppl. 115, 1-378 (1994).
35. Onsager, L.: Crystal statistics I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117-149 (1944).
36. Parisi, G.: Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. 43, 1754-1756 (1979).
37. Parisi, G., Slanina, F.: Loop expansion around the Bethe-Peierls approximation for lattice models. J. Stat. Mech. Theory Exp. L02003 (2006).
38. Peierls, R.: On Ising's model of ferromagnetism. Proc. Camb. Philol. Soc. 32, 477-481 (1936).
39. Pelizzola, A.: Cluster variation method in statistical physics and probabilistic graphical models. J. Phys. A, Math. Gen. 38, R309-R339 (2005).
40. Rizzo, T., Lage-Castellanos, A., Mulet, R., Ricci-Tersenghi, F.: Replica cluster variational method. J. Stat. Phys. 139, 375-416 (2010).
41. Rizzo, T., Wemmenhove, B., Kappen, H. J.: Cavity approximation for graphical models. Phys. Rev. E 76, 011102 (2007).
42. Rota, G. C.: On the foundations of combinatorial theory I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheor. 2, 340-368 (1964).
43. Roudi, Y., Hertz, J.: Mean field theory for nonequilibrium network reconstruction. Phys. Rev. Lett. 106, 048702 (2011).
44. Saul, L., Kardar, M.: Exact integer algorithm for the two-dimensional ±j Ising spin glass. Phys. Rev. E 48, R3221-R3224 (1993).
45. Sherrington, D., Kirkpatrick, S.: Solvable model of a spin-glass. Phys. Rev. Lett. 35, 1792-1796 (1975).
46. Suzuki, M., Hu, X., Hatano, N., Katori, M., Minami, K., Lipowski, A., Nonomura, Y.: Coherent Anomaly Method: Mean Field, Fluctuations and Systematics. World Scientific, Singapore (1995).
47. Tanaka, T.: Statistical mechanics of CDMA multiuser demodulation. Europhys. Lett. 54, 540-546 (2001).
48. Thomas, C. K., Huse, D. A., Middleton, A. A.: Zero- and low-temperature behavior of the two-dimensional ±j Ising spin glass. Phys. Rev. Lett. 107, 047203 (2011).
49. Toulouse, G.: Theory of the frustration effect in spin glasses: I. Commun. Phys. 2, 115-119 (1977).
50. Viana, L., Bray, A. J.: Phase diagrams for dilute spin glasses. J. Phys. C, Solid State Phys. 18, 3037-3051 (1985).
51. Weigt, M., White, R. A., Szurmant, H., Hoch, J. A., Hwa, T.: Identification of direct residue contacts in protein-protein interaction by message-passing. Proc. Natl. Acad. Sci. USA 106, 67-72 (2009).
52. Xiao, J. Q., Zhou, H.: Partition function loop series for a general graphical model: free-energy corrections and message-passing equations. J. Phys. A, Math. Theor. 44, 425001 (2011).
53. Yedidia, J. S., Freeman, W. T., Weiss, Y.: Constructing free-energy approximations and generalized belief-propagation algorithms. IEEE Trans. Inf. Theory 51, 2282-2312 (2005).
54. Zdeborová, L.: Statistical physics of hard optimization problems. Acta Phys. Slovaca 59, 169-303 (2009).
55. Zhou, H.: Boltzmann distribution of free energies in a finite-connectivity spin-glass system and the cavity approach. Front. Phys. China 2, 238-250 (2007).
56. Zhou, H., Wang, C., Xiao, J. Q., Bi, Z.: Partition function expansion on region-graphs and message-passing equations. J. Stat. Mech. Theory Exp. L12001 (2011).