Metastable states in the hierarchical Dyson model drive parallel processing in the hierarchical Hopfield network

In this paper, we introduce and investigate the statistical mechanics of hierarchical neural networks. First, we approach these systems ála Mattis, by thinking of the Dyson model as a single-pattern hierarchical neural network. We also discuss the stability of different retrievable states as predicted by the related self-consistencies obtained both from a mean-field bound and from a bound that bypasses the mean-field limitation. The latter is worked out by properly reabsorbing the magnetization fluctuations related to higher levels of the hierarchy into effective fields for the lower levels. Remarkably, mixing Amits ansatz technique for selecting candidate-retrievable states with the interpolation procedure for solving for the free energy of these states, we prove that, due to gauge symmetry, the Dyson model accomplishes both serial and parallel processing. We extend this scenario to multiple stored patterns by implementing the Hebb prescription for learning within the couplings. This results in Hopfield-like networks constrained on a hierarchical topology, for which, by restricting to the low-storage regime where the number of patterns grows at its most logarithmical with the amount of neurons, we prove the existence of the thermodynamic limit for the free energy, and we give an explicit expression of its mean-field bound and of its related improved bound. We studied the resulting self-consistencies for the Mattis magnetizations, which act as order parameters, are studied and the stability of solutions is analyzed to get a picture of the overall retrieval capabilities of the system according to both mean-field and non-mean-field scenarios. Our main finding is that embedding the Hebbian rule on a hierarchical topology allows the network to accomplish both serial and parallel processing. By tuning the level of fast noise affecting it or triggering the decay of the interactions with the distance among neurons, the system may switch from sequential retrieval to multitasking features, and vice versa. However, since these multitasking capabilities are basically due to the vanishing 'dialogue' between spins at long distance, this effective penury of links strongly penalizes the networks capacity, with results bounded by low storage.