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Understanding the glassy nature of neural networks is pivotal both fortheoretical and computational advances in Machine Learning and TheoreticalArtificial Intelligence. Keeping the focus on dense associative Hebbian neuralnetworks, the purpose of this paper is two-fold: at first we develop rigorousmathematical approaches to address properly a statistical mechanical picture ofthe phenomenon of {\em replica symmetry breaking} (RSB) in these networks, then-- deepening results stemmed via these routes -- we aim to inspect the {\emglassiness} that they hide. In particular, regarding the methodology, weprovide two techniques: the former is an adaptation of the transport PDE to thecase, while the latter is an extension of Guerra's interpolation breakthrough.Beyond coherence among the results, either in replica symmetric and in theone-step replica symmetry breaking level of description, we prove the Gardner'spicture and we identify the maximal storage capacity by a ground-state analysisin the Baldi-Venkatesh high-storage regime.  In the second part of the paper we investigate the glassy structure of thesenetworks: in contrast with the replica symmetric scenario (RS), RSB actuallystabilizes the spin-glass phase. We report huge differences w.r.t. the standardpairwise Hopfield limit: in particular, it is known that it is possible toexpress the free energy of the Hopfield neural network as a linear combinationof the free energies of an hard spin glass (i.e. the Sherrington-Kirkpatrickmodel) and a soft spin glass (the Gaussian or "spherical" model). This is nolonger true when interactions are more than pairwise (whatever the level ofdescription, RS or RSB): for dense networks solely the free energy of the hardspin glass survives, proving a huge diversity in the underlying glassiness ofassociative neural networks.

Replica symmetry breaking in dense neural networks