Ricerc@Sapienza

In recent decades, statistical mechanics of disordered systems (mainly spin-glasses) has become one of the main tools to investigate complex systems, probably due to the celebrated Replica Symmetry Breaking scheme of Parisi Theory and its deep implications. In this work we consider the analog bipartite spin-glass (or real-valued restricted Boltzmann machine in a neural-network jargon), whose variables (those quenched as well as those dynamical) share standard Gaussian distributions.

In this work, we introduce and investigate the properties of the “relativistic” Hopfield model endowed with temporally correlated patterns.
First, we review the “relativistic” Hopfield model and we briefly describe the experimental evidence underlying correlation among patterns.
Then, we face the study of the resulting model exploiting statistical-mechanics tools in a low-load regime. More precisely, we prove the
existence of the thermodynamic limit of the related free energy and we derive the self-consistence equations for its order parameters. These

In this paper we adapt the broken replica interpolation technique (developed by Francesco Guerra to deal with the Sherrington-Kirkpatrick model, namely a pairwise mean-field spin-glass whose couplings are i.i.d. standard Gaussian variables) in order to work also with the Hopfield model (i.e. a pairwise mean-field neural-network whose couplings are drawn according to Hebb's learning rule): This is accomplished by grafting Guerra's telescopic averages on the transport equation technique, recently developed by some of the authors.

We consider a three-layer Sejnowski machine and show that features learnt via contrastive divergence have a dual representation as patterns in a dense associative memory of order P = 4. The latter is known to be able to Hebbian store an amount of patterns scaling as NP -1, where N denotes the number of constituting binary neurons interacting P wisely.

We develop a real space renormalization group analysis of disordered models of glasses, in particular of the spin models at the origin of the random first-order transition theory. We find three fixed points, respectively, associated with the liquid state, with the critical behavior, and with the glass state. The latter two are zero-temperature ones; this provides a natural explanation of the growth of effective activation energy scale and the concomitant huge increase of relaxation time approaching the glass transition.

For every physical model defined on a generic graph or factor graph, the Bethe M-layer construction allows building a different model for which the Bethe approximation is exact in the large M limit, and coincides with the original model for M=1. The 1/M perturbative series is then expressed by a diagrammatic loop expansion in terms of so-called fat diagrams. Our motivation is to study some important second-order phase transitions that do exist on the Bethe lattice, but are either qualitatively different or absent in the corresponding fully connected case.