Recent observations of the light component of the cosmic-ray spectrum have revealed unexpected features that motivate further and more precise measurements up to the highest energies. The Dark Matter Particle Explorer (DAMPE) is a satellite-based cosmic-ray experiment that is operational since December 2015, continuously collecting data on high-energy cosmic particles with very good statistics, energy resolution, and particle identification capabilities. In this work, the latest measurements of the energy spectrum of proton+helium in the energy range from 46 GeV to 316 TeV are presented. Among the most distinctive features of the spectrum, a spectral hardening at ∼600 GeV has been observed, along with a softening at ∼29 TeV measured with a 6.6σ significance. Moreover, by measuring the energy spectrum up to 316 TeV, a strong link is established between space-and ground-based experiments, also suggesting the presence of a second hardening at ∼150 TeV. * https://geant4.web.cern.ch/node/302 † https://web.ikp.kit.edu/rulrich/crmc.html

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 N P −1 , where N denotes the number of constituting binary neurons interacting P -wisely. We also prove that, by keeping the dense associative network far from the saturation regime (namely, allowing for a number of patterns scaling only linearly with N , while P > 2) such a system is able to perform pattern recognition far below the standard signal-to-noise threshold. In particular, a network with P = 4 is able to retrieve information whose intensity is O(1) even in the presence of a noise O( √ N ) in the large N limit. This striking skill stems from a redundancy representation of patterns -which is afforded given the (relatively) low-load information storage -and it contributes to explain the impressive abilities in pattern recognition exhibited by new-generation neural networks. The whole theory is developed rigorously, at the replica symmetric level of approximation, and corroborated by signal-to-noise analysis and Monte Carlo simulations. arXiv:1911.12689v1 [cond-mat.dis-nn]

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). Remarkably 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 (where the number of storable patterns exactly equals the number of neurons the network is built of).

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In neural network's Literature, "Hebbian learning" traditionally refers to the procedure by which the Hopfield model and its generalizations "store" archetypes (i.e., definite patterns that are experienced just once to form the synaptic matrix). However, the term "learning" in Machine Learning refers to the ability of the machine to extract features from the supplied dataset (e.g., made of blurred examples of these archetypes), in order to make its own representation of the unavailable archetypes. Here, given a sample of examples, we define a supervised learning protocol based on Hebb's rule and by which the Hopfield network can infer the archetypes. By an analytical inspection, we detect the correct control parameters (including size and quality of the dataset) that tune the system performance and we depict its phase diagram. We also prove that, for structureless datasets, the Hopfield model equipped with this supervised learning rule is equivalent to a restricted Boltzmann machine and this suggests an optimal and interpretable training routine. Finally, this approach is generalized to structured datasets: we highlight a ultrametric-like organization (reminiscent of replica-symmetry-breaking) in the analyzed datasets and, consequently, we introduce an additional "broken-replica hidden layer" for its (partial) disentanglement, which is shown to improve MNIST classification from 75\% to 95\%, and to offer a new perspective on deep architectures.

We consider a multi-layer Sherrington-Kirkpatrick spin-glass as a model for deep restricted Boltzmann machines and we solve for its quenched free energy, in the thermodynamic limit and allowing for a first step of replica symmetry breaking. This result is accomplished rigorously exploiting interpolating techniques and recovering the expression already known for the replica-symmetry case. Further, we drop the restriction constraint by introducing intra-layer connections among spins and we show that the resulting system can be mapped into a modular Hopfield network, which is also addressed rigorously via interpolating techniques up to the first step of replica symmetry breaking.

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