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In this paper we introduce and investigate the statistical mechanics ofhierarchical neural networks: First, we approach these systems \`a la Mattis,by thinking at the Dyson model as a single-pattern hierarchical neural networkand we discuss the stability of different retrievable states as predicted bythe related self-consistencies obtained from a mean-field bound and from abound that bypasses the mean-field limitation. The latter is worked out byproperly reabsorbing fluctuations of the magnetization related to higher levelsof the hierarchy into effective fields for the lower levels. Remarkably, mixingAmit's ansatz technique (to select candidate retrievable states) with theinterpolation procedure (to solve for the free energy of these states) we provethat (due to gauge symmetry) the Dyson model accomplishes both serial andparallel processing. One step forward, we extend this scenario toward multiplestored patterns by implementing the Hebb prescription for learning within thecouplings. This results in an Hopfield-like networks constrained on ahierarchical topology, for which, restricting to the low storage regime (wherethe number of patterns grows at most logarithmical with the amount of neurons),we prove the existence of the thermodynamic limit for the free energy and wegive an explicit expression of its mean field bound and of the related improvedbound

Meta-stable states in the hierarchical Dyson model drive parallel processing in the hierarchical Hopfield network