Dottorato in MATEMATICA

Titolo della tesi: Mathematical models and methods for neural networks that learn and retrive

The studies of disordered systems, such as spin glass systems has played a pivotal role in understanding Artificial Neural Networks and information processing. The Curie-Weiss and Sherrington-Kirkpatrick models can be considered as two paradigmatic statistical mechanics models to study neural networks from a mathematical-physical perspective. In this thesis we first introduce to these models to then move to a detailed description of the two key models of artificial intelligence, i.e. the Restricted Boltzmann Machine (RBM) and the Hopfield model (HN). We will show that, interestingly, although they represent two distinct cognitive processes (i.e. learning and information retrieval), are nothing more than two sides of the same coin. In fact, once the RBM is trained, in its future usage it will behave as an HN for pattern recognition. After this introductory part, we will walk the path of my works concerning the extensions of these classical models. Initially, a "relativistic" version of the HN, where we introduced a mechanical analogy between the HN free energy and the Hamilton-Jacobi equation describing a fictitious single particle motion is presented. Next, a more theoretical result regarding p-spin HN model is studied. In particular, we will present an analytical methods, based on nonlinear PDEs, to investigate their functioning and proving differential identities involving macroscopic observables useful for a qualitative and quantitative analysis of the system. We will then show that the duality between the HN and the RBM models can be extended to the case of biased patterns (i.e. patterns which display an unbalanced count of positive neurons/pixels) introducing in the Hamiltonian a constraint parameter for the bias correction. Finally, we will see that the Pavlovian correlations among concepts and the Hebb's correlations among pairs of adjacent neurons are intrinsically connected and that, in particular, Pavlov mechanism generates the synaptic weights of the Hebbian kernel.