Gibbs measures associated with the fully visible Boltzmann machine
In this work, we have undertaken a rigorous study of Boltzmann machines and the associated
Gibbs measures they induce. Beginning from the machine’s energy function, we have detailed
the construction of the corresponding Gibbs measures and derived the requisite consistency
(Kolmogorov) conditions. We have further identified precise criteria under which these consistency
conditions hold and have evaluated their implications for the coherence and validity of statistical
models built upon Boltzmann-machine architectures.
1. Bleher P.M. and Ganikhodjaev N.N., On pure phases of the Ising model on the Bethe lattice. Theor.
Probab. Appl. 35 (1990), 216-227.
2. Bleher P.M., Ruiz J. and Zagrebnov V.A., On the purity of the limiting Gibbs state for the Ising model
on the Bethe lattice. Journ. Statist. Phys. 79 (1995), 473-482.
3. Rozikov U.A.: Gibbs Measure on Cayley trees. (2013). World Scientific Publishing Co.Pte. Ltd.
4. Eshkabilov Yu.Kh., Haydarov F.H., Rozikov U.A.: Non-uniqueness of Gibbs Measure for Models with
Uncountable Set of Spin Values on a Cayley Tree. J. Stat. Phys. 147(2012),779-794.
5. Eshkabilov Yu.Kh, Haydarov F.H., Rozikov U.A.: Uniqueness of Gibbs Measure for Models With
Uncountable Set of Spin Values on a Cayley Tree. Math. Phys. Anal. Geom. 16 (2013), 1-17.
6. Eshkabilov Yu.Kh., Rozikov U.A. and Botirov G.I.: Phase Transitions for a Model with Uncountable Set
of Spin Values on a Cayley Tree. Lobachevskii Journal of Mathematics, 34 (2013), No.3, 256-263.
7. Ganikhodjaev N.N. and Rozikov U.A. Description of periodic extreme Gibbs measures of some lattice
model on the Cayley tree. Theor. and Math. Phys. 111 (1997), 480-486.
8. Azencott R. Synchronous Boltzmann Machines and Gibbs Fields: Learning Algorithms. In: Soulie, F.F.,
Herault, J. (eds) Neurocomputing. NATO ASI Series, vol 68. Springer, Berlin, Heidelberg, (1990), 51-62.
9. Ganikhodjaev N.N. and Rozikov U.A. On Ising model with four competing interactions on Cayley tree.
Math. Phys. Anal. Geom. 12 (2009), 141-156.
10. Jahnel B., Christof K., Botirov G. Phase transition and critical value of nearest-neighbor system with
uncountable local state space on Cayley tree. Math. Phys. Anal. Geom. 17 (2014) 323-331.
11. Preston C.: Gibbs states on countable sets (Cambridge University Press, London 1974).
12. Rozikov U.A. Partition structures of the Cayley tree and applications for describing periodic Gibbs
distributions. Theor. and Math. Phys. 112 (1997), 929-933.
13. Rozikov U.A. and Eshkabilov Yu.Kh.: On models with uncountable set of spin values on a Cayley tree:
Integral equations. Math. Phys. Anal. Geom. 13 (2010), 275-286.
14. Rozikov U. A., Haydarov F. H. Periodic Gibbs measures for models with uncountable set of spin values
on a Cayley tree. I.D.A.Q.P. 18 (2015), 1-22.
15. Sinai Ya.G.: Theory of phase transitions: Rigorous Results (Pergamon, Oxford, 1982).
16. Pigorov S.A., Sinai Ya.G.: Phase diagrams of classical lattice systems (Russian). Theor. and Math. Phys.
25 (1975), 358-369.
17. Pigorov S.A., Sinai Ya.G.: Phase diagrams of classical lattice systems. Continuation (Russian). Theor. and
Math. Phys. 26 (1976), 61-76.
18. Kotecky R. and Shlosman S.B.: First-order phase transition in large entropy lattice models. Commun.
Math. Phys. 83 (1982), 493-515.
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