Chaotic dynamics of high-order neural networks

The dynamics of an extremely diluted neural network with high-order synapses acting as corrections to the Hopfield model is investigated. The learning rules for the high-order connections contain mixing of memories, different from all the previous generalizations of the Hopfield model. The dynamics may display fixed points or periodic and chaotic orbits, depending on the weight of the high-order connections , the noise levelT, and the network load, defined as the ratio between the number of stored patterns and the mean connectivity per neuron, =P/C. As in the related fully connected case, there is an optimal value of the weight that improves the storage capacity of the system (the capacity diverges).     

Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows

We prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form, in dimension three. This is the first time exponential decay of correlations is proved for continuous-time dynamics with singularities on a manifold. Our proof combines the second author’s version (Liverani in Ann Math 159:1275–1312, 2004) of Dolgopyat’s estimates for contact flows and the first author’s work with Gouëzel (J Mod Dyn 4:91–137, 2010) on piecewise hyperbolic discrete-time dynamics.     

Thermodynamic limit of theq-state Potts-Hopfield model with infinitely many patterns

We prove the almost sure convergence of the free energy and of the overlap order parameters in aq-state version of the Hopfield neural network model. We compute explicitly these limits for all temperatures different from some critical value. The number of stored patterns is allowed to grow with the size of the systemN like (/lnq) lnN. We study the limiting behavior of the extremal states of the model that are the measures induced on the Gibbs measures by the overlap parameters.     

The free energy of a class of Hopfield models

We show that in the limitp ,N 0,=p/N 0 the limit free energy of the Hopfield model equals in probability the Curie-Weiss free energy. We prove also that the free energy of the Hopfield model is self-averaging for any finite .     

The Kac Version of the Sherrington–Kirkpatrick Model at High Temperatures

We study the Kac version of the Sherrington–Kirkpatrick (SK) model of a spin glass, i.e., a spin glass with long- but finite-range interaction on and Gaussian mean zero couplings. We prove that for all < 1, the free energy of this model converges to that of the SK model as the range of the interaction tends to infinity. Moreover, we prove that for all temperatures for which the infinite-volume Gibbs state is unique, the free energy scaled by the square root of the volume converges to a Gaussian with variance c _, , where ^–1 is the range of the interaction. Moreover, at least for almost all values of , this variance tends to zero as goes to zero, the value in the SK model. We interpret our finding as a weak indication that at least at high temperatures, the SK model can be seen as a reasonable asymptotic model for lattice spin glasses.     

Exponential Convergence to Equilibrium for the Homogeneous Boltzmann Equation for Hard Potentials Without Cut-Off

This paper deals with the long time behavior of solutions to the spatially homogeneous Boltzmann equation. The interactions considered are the so-called (non cut-off with a moderate angular singularity and non mollified) hard potentials. We prove an exponential in time convergence towards the equilibrium, improving results of Villani (Commun Math Phys 234(3): 455–490, 2003) where a polynomial decay to equilibrium is proven. The basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a \(L^1\) space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by Gualdani et al. (http://hal.archives-ouvertes.fr/ccsd-00495786, 2013). We then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish.     

Computational complexity,learning rules and storage capacities: A Monte Carlo study for the binary perceptron

We examine the storage capacity for the binary perceptron using simulated annealing. In particular, we clarify the connection between the computational complexity of learning algorithms and the attained storage capacity. From finite-size studies we obtain a critical storage capacity, _c ()=0.8331±0.0016, in good agreement with the replica analysis of Krauth and Mézard. However, we demonstrate that a polynomial time cooling schedule yields a vanishing storage capacity in the thermodynamic limit as predicted by the dynamical theory of Horner. Nonetheless, we show these two results may be reconciled by explicitly verifying that the learning problem for the binary perceptron is NP-complete. This investigation has been made possible by the development of an accelerated annealing algorithm.     

A multineuron interaction model for neural networks

A multineuron interaction model (RS model) with an energy function given by the product of the squared distances in phase space between the state of the net and the stored patterns is studied in detail within a mean-field approach. Two limits are considered: when the patterns and antipatterns are stored (as in the Hopfield model), PAS case, and when only the patterns are taken into account, OPS case. TheT=0 solutions for the proper memories are exactly obtained for all finite values of, as a consequence of the energy function: whenever one of the overlaps is exactly one the corresponding equations decouple and no configuration average is required. Special interest is focused on the OPS situation, which presents a peculiar phase space topology. On the other hand, the PAS configuration recovers the Hopfield model in the appropriate limit, while keeping associative memory abilities far beyond the critical values of other models when the full Hamiltonian is considered.     

Resonant decay of a two state atom interacting with a massless non-relativistic quantised scalar field

We consider a model Hamiltonian derived from the interaction of an atom with a non-relativistic massless quantized field. The model atom has two states, and the interaction is linear in the field operator. We do not make the rotating wave approximation and there is no infrared cutoff. We prove that the excited state of the atom with no photons present decays at an approximately exponential rate in accordance with the predictions of time dependent perturbation theory. The proof requires some analyticity and regularity assumptions on the interaction between atom and field. These imply in particular that the interaction goes to zero at least as fast ask ², ask0, wherek is the photon momentum.     

From granules to bulk superconductors using Richardson-Gaudin equations

We provide a compact expression of the ground-state energy of N-Cooper pairs valid from small to large sample volumes, as checked by numerically solving Richardson-Gaudin equations which give the exact eigenstates of BCS superconductors. This expression contains a contribution linear in the potential amplitude, dominant for small samples, and an exponential contribution dominant when the number of states available for pairing gets larger than a material-dependent threshold independent from sample size. These “available states” are the states feeling the BCS potential, reduced by the Pauli exclusion principle through a “moth-eaten effect” which comes from the composite boson nature of Cooper pairs. This work also presents an elegant derivation of the N-Cooper pair energy obtained recently, which makes use of the roots of the degree-N Hermite polynomial.     

The Kac Model Coupled to a Thermostat

In this paper we study a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature \(\beta \) . The system admits the canonical distribution at inverse temperature \(\beta \) as the unique equilibrium state. We prove that any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one particle marginal, in the large system limit, satisfies an effective Boltzmann-type equation.     

Truncated Gamow functions, α-decay and the exponential law

For a quantum mechanical two-bodys-wave resonance we prove that the evolution of square integrable approximations of the Gamow function is outgoing and exponentially damped. An error estimate is given in terms of resonance energy and explicity. We obtain the Breit-Wigner form. The results are used in an -decay model to prove general validity of the exponential decay law for periods of several lifetimes.     

External Stimuli on Neural Networks: Analytical and Numerical Approaches

Based on the behavior of living beings, which react mostly to external stimuli, we introduce a neural-network model that uses external patterns as a fundamental tool for the process of recognition. In this proposal, external stimuli appear as an additional field, and basins of attraction, representing memories, arise in accordance with this new field. This is in contrast to the more-common attractor neural networks, where memories are attractors inside well-defined basins of attraction. We show that this procedure considerably increases the storage capabilities of the neural network; this property is illustrated by the standard Hopfield model, which reveals that the recognition capacity of our model may be enlarged, typically, by a factor 102. The primary challenge here consists in calibrating the influence of the external stimulus, in order to attenuate the noise generated by memories that are not correlated with the external pattern. The system is analyzed primarily through numerical simulations. However, since there is the possibility of performing analytical calculations for the Hopfield model, the agreement between these two approaches can be tested—matching results are indicated in some cases. We also show that the present proposal exhibits a crucial attribute of living beings, which concerns their ability to react promptly to changes in the external environment. Additionally, we illustrate that this new approach may significantly enlarge the recognition capacity of neural networks in various situations; with correlated and non-correlated memories, as well as diluted, symmetric, or asymmetric interactions (synapses). This demonstrates that it can be implemented easily on a wide diversity of models.     

Flow version of statistical neurodynamics for oscillator neural networks

We consider a neural network of Stuart–Landau oscillators as an associative memory. This oscillator network with N$N$ elements is a system of an N$N$-dimensional differential equation, works as an attractor neural network, and is expected to have no Lyapunov functions. Therefore, the technique of equilibrium statistical physics is not applicable to the study of this system in the thermodynamic limit. However, the simplicity of this system allows us to extend statistical neurodynamics [S. Amari, K. Maginu, Neural Netw. 1 (1988) 63–73], which was originally developed to analyse the discrete time evolution of the Hopfield model, into the version for continuous time evolution. We have developed and attempted to apply this method in the analysis of the phase transition of our model network.     

From Cycle Rooted Spanning Forests to the Critical Ising Model: an Explicit Construction

Fisher established an explicit correspondence between the 2-dimensional Ising model defined on a graph G and the dimer model defined on a decorated version ${\mathcal{G}}$ of this graph (Fisher in J Math Phys 7:1776–1781, 1966). In this paper we explicitly relate the dimer model associated to the critical Ising model and critical cycle rooted spanning forests (CRSFs). This relation is established through characteristic polynomials, whose definition only depends on the respective fundamental domains, and which encode the combinatorics of the model. We first show a matrix-tree type theorem establishing that the dimer characteristic polynomial counts CRSFs of the decorated fundamental domain ${\mathcal{G}_1}$ . Our main result consists in explicitly constructing CRSFs of ${\mathcal{G}_1}$ counted by the dimer characteristic polynomial, from CRSFs of G ₁, where edges are assigned Kenyon’s critical weight function (Kenyon in Invent Math 150(2):409–439, 2002); thus proving a relation on the level of configurations between two well known 2-dimensional critical models.     

Stable exponential cosmological solutions with two factor spaces in the Einstein–Gauss–Bonnet model with a $$\Lambda $$-term

We study D-dimensional Einstein–Gauss–Bonnet gravitational model including the Gauss–Bonnet term and the cosmological term \(\Lambda \). We find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters \(H >0\) and h, corresponding to factor spaces of dimensions \(m >2\) and \(l > 2\), respectively. These solutions contain a fine-tuned \(\Lambda = \Lambda (x, m, l, \alpha )\), which depends upon the ratio \(h/H = x\), dimensions of factor spaces m and l, and the ratio \(\alpha = \alpha _2/\alpha _1\) of two constants (\(\alpha _2\) and \(\alpha _1\)) of the model. The master equation \(\Lambda (x, m, l,\alpha ) = \Lambda \) is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals. The explicit solution for \(m = l\) is presented in “Appendix”. Imposing certain restrictions on x, we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. We also consider a subclass of solutions with small enough variation of the effective gravitational constant G and show the stability of all solutions from this subclass.     

Scalar Domination and Normal Fluctuations in N-Vector Quantum Anharmonic Crystals

A lattice model of N-dimensional quantum anharmonic oscillators with a polynomial anharmonicity and a ferroelectric pair interaction is considered. For arbitrary N , correlation inequalities, showing that the temperature Green functions of this model are dominated by the corresponding Green functions of the scalar (N=1) model, are proven. These inequalities are then used to prove that the fluctuations of displacements of particles remain normal at all temperatures provided the model parameters obey a certain condition. In particular this means that the smallest distance between the energy levels of the corresponding one-dimensional isolated oscillator should be large enough or its mass should be small enough.     

Approximation Algorithms for Two-State Anti-Ferromagnetic Spin Systems on Bounded Degree Graphs

We show that for the anti-ferromagnetic Ising model on the Bethe lattice, weak spatial mixing implies strong spatial mixing. As a by-product of our analysis, we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti-ferromagnetic Ising model (with non-zero external field) on the Bethe lattice. Following a method due to Weitz [15], we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of degree at most  $d$ , throughout the uniqueness region of the Gibbs measure on the infinite $d$ -regular tree. By a standard correspondence, our results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints. Subsequent to a preliminary version of this paper, Sly and Sun [13] have shown that our results are optimal in the sense that, under standard complexity theoretic assumptions, there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree  $d$ for parameters outside the uniqueness region. Taken together, the results of [13] and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems.     

Rigorous bounds on the storage capacity of the dilute Hopfield model

We study a neural network model consisting ofN neurons where a dendritic connection between each pair of neurons exists with probabilityp and is absent with probability 1-p. For the Hopfield Hamiltonian on such a network, we prove that ifp c[(lnN)/N]^1/2, the model can store at leastm= _cpN patterns, where _c 0.027 ifc 3 and decreases proportional to 1/(–lnc) forc small. This generalizes the results of Newman for the standard Hopfield model.     

Rigorous Results for Hierarchical Models of Structural Glasses

We consider two non-mean-field models of structural glasses built on a hierarchical lattice. First, we consider a hierarchical version of the random energy model, and we prove the existence of the thermodynamic limit and self-averaging of the free energy. Furthermore, we prove that the infinite-volume entropy is positive in a high-temperature region bounded from below, thus providing an upper bound on the Kauzmann critical temperature. In addition, we show how to improve this bound by leveraging the hierarchical structure of the model. Finally, we introduce a hierarchical version of the \(p\) -spin model of a structural glass, and we prove the existence of the thermodynamic limit and self-averaging of the free energy.