Dense Hebbian neural networks: a replica symmetric picture of unsupervised learning

We consider dense, associative neural-networks trained with no supervision and we investigate their computational capabilities analytically, via statistical-mechanics tools, and numerically, via Monte Carlo simulations. In particular, we obtain a phase diagram summarizing their performance as a function of the control parameters (e.g. quality and quantity of the training dataset, network storage, noise) that is valid in the limit of large network size and structureless datasets. Moreover, we establish a bridge between macroscopic observables standardly used in statistical mechanics and loss functions typically used in the machine learning. As technical remarks, from the analytical side, we extend Guerra's interpolation to tackle the non-Gaussian distributions involved in the post-synaptic potentials while, from the computational counterpart, we insert Plefka's approximation in the Monte Carlo scheme, to speed up the evaluation of the synaptic tensor, overall obtaining a novel and broad approach to investigate unsupervised learning in neural networks, beyond the shallow limit.