We consider restricted Boltzmann machine (RBMs) trained over an unstructured dataset made of blurred copies of definite but unavailable ``archetypes'' and we show that there exists a critical sample size beyond which the RBM can learn archetypes, namely the machine can successfully play as a generative model or as a classifier, according to the operational routine. In general, assessing a critical sample size (possibly in relation to the quality of the dataset) is still an open problem in machine learning. Here, restricting to the random theory, where shallow networks suffice and the grand-mother cell scenario is correct, we leverage the formal equivalence between RBMs and Hopfield networks, to obtain a phase diagram for both the neural architectures which highlights regions, in the space of the control parameters (i.e., number of archetypes, number of neurons, size and quality of the training set), where learning can be accomplished. Our investigations are led by analytical methods based on the statistical-mechanics of disordered systems and results are further corroborated by extensive Monte Carlo simulations.

Recently a daily routine for associative neural networks has been proposed: the network Hebbian-learns during the awake state (thus behaving as a standard Hopfield model), then, during its sleep state, optimizing information storage, it consolidates pure patterns and removes spurious ones: this forces the synaptic matrix to collapse to the projector one (ultimately approaching the Kanter-Sompolinksy model). This procedure keeps the learning Hebbian-based (a biological must) but, by taking advantage of a (properly stylized) sleep phase, still reaches the maximal critical capacity (for symmetric interactions). So far this emerging picture (as well as the bulk of papers on unlearning techniques) was supported solely by mathematically-challenging routes, e.g. mainly replica-trick analysis and numerical simulations: here we rely extensively on Guerra's interpolation techniques developed for neural networks and, in particular, we extend the generalized stochastic stability approach to the case. Confining our description within the replica symmetric approximation (where the previous ones lie), the picture painted regarding this generalization (and the previously existing variations on theme) is here entirely confirmed. Further, still relying on Guerra's schemes, we develop a systematic fluctuation analysis to check where ergodicity is broken (an analysis entirely absent in previous investigations). We find that, as long as the network is awake, ergodicity is bounded by the Amit-Gutfreund-Sompolinsky critical line (as it should), but, as the network sleeps, sleeping destroys spin glass states by extending both the retrieval as well as the ergodic region: after an entire sleeping session the solely surviving regions are retrieval and ergodic ones and this allows the network to achieve the perfect retrieval regime (the number of storable patterns equals the number of neurons in the network).

We propose a modification of the cost function of the Hopfield model whose salient features shine in its Taylor expansion and result in more than pairwise interactions with alternate signs, suggesting a unified framework for handling both with deep learning and network pruning. In our analysis, we heavily rely on the Hamilton-Jacobi correspondence relating the statistical model with a mechanical system. In this picture, our model is nothing but the relativistic extension of the original Hopfield model (whose cost function is a quadratic form in the Mattis magnetization which mimics the non-relativistic Hamiltonian for a free particle). We focus on the low-storage regime and solve the model analytically by taking advantage of the mechanical analogy, thus obtaining a complete characterization of the free energy and the associated self-consistency equations in the thermodynamic limit. On the numerical side, we test the performances of our proposal with MC simulations, showing that the stability of spurious states (limiting the capabilities of the standard Hebbian construction) is sensibly reduced due to presence of unlearning contributions in this extended framework.

We study Generalised Restricted Boltzmann Machines with generic priors for units and weights, interpolating between Boolean and Gaussian variables. We present a complete analysis of the replica symmetric phase diagram of these systems, which can be regarded as Generalised Hopfield models. We underline the role of the retrieval phase for both inference and learning processes and we show that retrieval is robust for a large class of weight and unit priors, beyond the standard Hopfield scenario. Furthermore we show how the paramagnetic phase boundary is directly related to the optimal size of the training set necessary for good generalisation in a teacher-student scenario of unsupervised learning.

Restricted Boltzmann Machines are described by the Gibbs measure of a bipartite spin glass, which in turn corresponds to the one of a generalised Hopfield network. This equivalence allows us to characterise the state of these systems in terms of retrieval capabilities, both at low and high load. We study the paramagnetic-spin glass and the spin glass-retrieval phase transitions, as the pattern (i.e. weight) distribution and spin (i.e. unit) priors vary smoothly from Gaussian real variables to Boolean discrete variables. Our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard Hopfield model with Boolean patterns. The retrieval region is larger when the pattern entries and retrieval units get more peaked and, conversely, when the hidden units acquire a broader prior and therefore have a stronger response to high fields. Moreover, at low load retrieval always exists below some critical temperature, for every pattern distribution ranging from the Boolean to the Gaussian case.

A specific type of neural network, the Restricted Boltzmann Machine (RBM), is implemented for classification and feature detection in machine learning. RBM is characterized by separate layers of visible and hidden units, which are able to learn efficiently a generative model of the observed data. We study a "hybrid" version of RBM's, in which hidden units are analog and visible units are binary, and we show that thermodynamics of visible units are equivalent to those of a Hopfield network, in which the N visible units are the neurons and the P hidden units are the learned patterns. We apply the method of stochastic stability to derive the thermodynamics of the model, by considering a formal extension of this technique to the case of multiple sets of stored patterns, which may act as a benchmark for the study of correlated sets. Our results imply that simulating the dynamics of a Hopfield network, requiring the update of N neurons and the storage of N(N-1)/2 synapses, can be accomplished by a hybrid Boltzmann Machine, requiring the update of N+P neurons but the storage of only NP synapses. In addition, the well known glass transition of the Hopfield network has a counterpart in the Boltzmann Machine: It corresponds to an optimum criterion for selecting the relative sizes of the hidden and visible layers, resolving the trade-off between flexibility and generality of the model. The low storage phase of the Hopfield model corresponds to few hidden units and hence a overly constrained RBM, while the spin-glass phase (too many hidden units) corresponds to unconstrained RBM prone to overfitting of the observed data.