Abstract
We have learned in the first part of this book that there is a complete analogy between neural networks having symmetric synaptic efficacies w ij and a certain type of magnetic system characterized by a lattice of discrete spin variables s i = ±1. Such systems are known as Ising systems. If the spin-spin interaction, i.e. the coupling coefficients w ij , extend over large distances and take on irregular values one speaks of a spin glass. The peculiar properties of such spin glasses have caught the attention of physicists and have been studied closely during the last decade. In the following chapters we will make ample use of the results and the methods developed in the course of these investigations. Despite the close magnetic analogy, however, we will always keep in mind that we intend to describe neural networks.
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Notes
Modern introductions to the methods of statistical physics are, e.g., those by Feynman [Fe72] and Reichl [Re80].
We partly follow here the presentation of Hertz, Krogh, and Palmer [He91].
For a statistical treatment of asymmetric neural networks, see [C188].
The best-known example of such materials is window glass, which is a “liquid” with virtually infinite viscosity.
Experiments on the magnetic properties of spin glass materials have shown relaxation times of 105 seconds and more to reach the equilibrium state [Ko89b].
The replicas are introduced to facilitate the calculation of the quenched average of ln Z.
The ferromagnetic phase is the one most interesting for analogous neural network models of associative memory, since its ground state corresponds to a memorized pattern.
Actually, this region of the phase diagram is characterized by the coexistence of an infinite multitude of phases corresponding to the various ground states, and one faces the unfamiliar task of doing a statistical mechanics of phases!
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© 1995 Springer-Verlag Berlin Heidelberg
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Müller, B., Reinhardt, J., Strickland, M.T. (1995). Statistical Physics and Spin Glasses. In: Neural Networks. Physics of Neural Networks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57760-4_17
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DOI: https://doi.org/10.1007/978-3-642-57760-4_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60207-1
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