A specific type of the neural networks, the Restricted Boltzmann Machines (RBM), are implemented for classification and feature detection. They are 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 the evolution and the 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 machine, 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 only the storage of N P 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 very constrained RBM, while the spin-glass phase (too many hidden units) corresponds to an overly unconstrained RBM prone to overfitting of the observed data.

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We introduce a framework suitable for describing standard classification problems using the mathematical language of quantum states. In particular, we provide a one-to-one correspondence between real objects and pure density operators. This correspondence enables us: (1) to represent the nearest mean classifier (NMC) in terms of quantum objects, (2) to introduce a quantum-inspired version of the NMC called quantum classifier (QC). By comparing the QC with the NMC on different datasets, we show how the first classifier is able to provide additional information that can be beneficial on a classical computer with respect to the second classifier

The aim of this paper is to provide a quantum counterpart of the well known minimum-distance classifier named Nearest Mean Classifier (NMC). In particular, we refer to the following previous works: i) in [13] we have introduced a detailed quantum version of the NMC, named Quantum Nearest Mean Classifier (QNMC), for two-dimensional problems and we have proposed a generalization to abitrary dimensions; ii) in [12] the n-dimensional problem was analyzed in detail and a particular encoding for arbitrary n-feature vectors into density operators has been presented. In this paper, we introduce a new promizing encoding of arbitrary n-dimensional patterns into density operators, starting from the two-feature encoding provided in [13]. Further, unlike the NMC, the QNMC shows to be not invariant by rescaling the features of each pattern. This property allows us to introduce a free parameter whose variation provides, in some case, an improvement of the QNMC performance. We show experimental results where: i) the NMC and QNMC performances are compared on different datasets; ii) the effects of the non-invariance under uniform rescaling for the QNMC are investigated.

We face the problem of pattern classification by proposing a quantuminspired version of the widely used minimum distance classifier (i.e. the Nearest Mean Classifier (NMC)) already introduced in [31,33,28,27] and by applying this quantuminspired classifier in a biomedical context. In particular, we show and compare the NMC and our quantum model performance to solve a problem related to classify the probability of survival for patients affected by idiopathic pulmonary fibrosis (IPF).