Do Hopfield networks dream of stored patterns? A statistical–mechanical theory of dreaming in multidirectional associative memories

We introduce the Dreaming L-directional Associative Memory (DLAM), a multi-layer Hebbian architecture in which off-line dreaming and supervised heteroassociative coupling coexist within a single cost function, placing our approach within the framework of energy-based models (EBMs). The replica-symmetric free energy, derived via the Guerra interpolation scheme, yields self-consistency equations governing the order parameters across the control-parameter space. The effective local field decomposes into signal, intra-layer dreaming noise, and inter-layer noise. Dreaming improves retrieval by differentially attenuating high-eigenvalue interference modes of the empirical correlation matrix, suppressing inter-pattern crosstalk while preserving the signal. Dreaming and inter-layer coupling prove synergistic, opening retrieval regions unreachable by either mechanism alone, as confirmed by Monte Carlo simulations for L=3. Their interplay is most pronounced on pattern disentanglement: given a mixture state as input, the network splits the constituent patterns one-per-layer, recovering each modality-specific pattern from a common cue that simultaneously blends noisy evidence from all sensory channels. Phase diagrams are planar projections of the hyperspace (α, T, ρ, t)-where α is the storage load, T the fast-noise temperature, ρ the dataset entropy, and t the sleeping time. In the (ρ, t)-plane, the diagrams reveal a data-computation trade-off: off-line consolidation substitutes for additional training data, extending to heteroassociative architectures a phenomenon previously established for autoassociative networks. Enriching the standard Hopfield model with heteroassociativity and dreaming gives rise to EBMs capable of complex tasks beyond classical pattern recognition, contributing to a modern theory of neural information processing.