Phase diagrams for locally Hopfield neural networks in presence of correlated patterns

Stochastic recurrent neural networks have been successful in many applications [Hopfield, J. J and Tank, D, 1985], [ Wilson, G. V. and Pawley, G. S., 1988], [Kuo-Sheng Cheng, et al.] and their position in the literature is now quite strong, in particular in context of networks aimed at mimicking the thermodynamic behavior of complex physical systems, where a number of theoretical tools and motivations are available, originating from the area of statistical mechanics. As prominent examples of such networks one can invoke Hopfield-type content-addressed associative memories and Boltzmann machines, which are often interpreted within the framework of the spin-glass theory. Yet there are many unsolved questions in theories of such systems. Whereas a lot is known about the characteristics of Hopfield networks learning random independent patterns, less is known about their behavior in the presence of strong correlations, especially in non-zero temperature systems. In this paper we consider non-zero low temperature large-size locally Hopfield neural networks in finite loading regime, learning patterns admitting strong and regular correlation structure (chosen to be periodic for simplicity). Our aim being to find conditions for pattern stability and successful retrieval as depending on the imposed correlation structure, we propose to apply the so-called Pirogov-Sinai theory from statistical mechanics to describe the geometry of pattern stability regions. We check that this theory does match with the experimental data and we investigate some further interesting phenomena that occur in such systems.