Inverse melting and inverse freezing in a three-state spin-glass model with finite connectivity.

The phase diagrams of the three-state Ghatak-Sherrington spin-glass (or random Blume-Capel) model are obtained in mean-field theory with replica symmetry in order to study the effects of a ferromagnetic bias and finite random connectivity in which each spin is connected to a finite number of other spins. It is shown that inverse melting from a ferromagnetic to a low-temperature paramagnetic phase may appear for small but finite disorder and that inverse freezing appears for large disorder. There can also be a continuous inverse ferromagnetic to spin-glass transition.

Phase transitions in the three-state Ising spin-glass model with finite connectivity.

The statistical mechanics of a two-state Ising spin-glass model with finite random connectivity, in which each site is connected to a finite number of other sites, is extended in this work within the replica technique to study the phase transitions in the three-state Ghatak-Sherrington (or random Blume-Capel) model of a spin glass with a crystal-field term. The replica symmetry ansatz for the order function is expressed in terms of a two-dimensional effective-field distribution, which is determined numerically by means of a population dynamics procedure. Phase diagrams are obtained exhibiting phase boundaries that have a reentrance with both a continuous and a genuine first-order transition with a discontinuity in the entropy. This may be seen as "inverse freezing," which has been studied extensively lately, as a process either with or without exchange of latent heat.

Period-two cycles in a feedforward layered neural network model with symmetric sequence processing.

The effects of dominant sequential interactions are investigated in an exactly solvable feedforward layered neural network model of binary units and patterns near saturation in which the interaction consists of a Hebbian part and a symmetric sequential term. Phase diagrams of stationary states are obtained and a phase of cyclic correlated states of period two is found for a weak Hebbian term, independently of the number of condensed patterns c.

Pattern reconstruction and sequence processing in feed-forward layered neural networks near saturation.

The dynamics and the stationary states for the competition between pattern reconstruction and asymmetric sequence processing are studied here in an exactly solvable feed-forward layered neural network model of binary units and patterns near saturation. Earlier work by Coolen and Sherrington on a parallel dynamics far from saturation is extended here to account for finite stochastic noise due to a Hebbian and a sequential learning rule. Phase diagrams are obtained with stationary states and quasiperiodic nonstationary solutions. The relevant dependence of these diagrams and of the quasiperiodic solutions on the stochastic noise and on initial inputs for the overlaps is explicitly discussed.

Time evolution of the extremely diluted Blume-Emery-Griffiths neural network.

A study of the time evolution and a stability analysis of the phases in the extremely diluted Blume-Emery-Griffiths neural network model are shown to yield new phase diagrams in which fluctuation retrieval may drive pattern retrieval. It is shown that saddle-point solutions associated with fluctuation overlaps slow down the flow of the network states towards the retrieval fixed points. A comparison of the performance with other three-state networks is also presented.

Retrieval behavior and thermodynamic properties of symmetrically diluted Q-Ising neural networks.

The retrieval behavior and thermodynamic properties of symmetrically diluted Q-Ising neural networks are derived and studied in replica-symmetric mean-field theory generalizing earlier works on either the fully connected or the symmetrical extremely diluted network. Capacity-gain parameter phase diagrams are obtained for the Q=3, Q=4, and Q=infinity state networks with uniformly distributed patterns of low activity in order to search for the effects of a gradual dilution of the synapses. It is shown that enlarged regions of continuous changeover into a region of optimal performance are obtained for finite stochastic noise and small but finite connectivity. The de Almeida-Thouless lines of stability are obtained for arbitrary connectivity, and the resulting phase diagrams are used to draw conclusions on the behavior of symmetrically diluted networks with other pattern distributions of either high or low activity.

Categorization in the symmetrically dilute Hopfield network.

A symmetrically dilute Hopfield model with a Hebbian learning rule is used to study the effects of gradual dilution and of synaptic noise on the categorization ability of an attractor neural network with hierarchically correlated patterns in a two-level structure of ancestors and descendants. Categorization consists in recognizing the ancestors when the network has been trained exclusively with the descendants. We consider a macroscopic number of ancestors, each with a finite number of descendants, and take into account the stochastic noise produced by the former in an equilibrium study of the network, by means of replica-symmetric mean-field theory. Phase diagrams are obtained that exhibit a categorization, a spin-glass, and a paramagnetic phase, as well as the dependence of the order parameters on the relevant quantities. The de Almeida-Thouless lines that limit the validity of the replica-symmetric results are also obtained. It is shown that gradual dilution increases considerably the region where a stable categorization phase may be found.

Categorization in fully connected multistate neural network models.

The categorization ability of fully connected neural network models, with either discrete or continuous Q-state units, is studied in this work in replica symmetric mean-field theory. Hierarchically correlated multistate patterns in a two level structure of ancestors and descendents (examples) are embedded in the network and the categorization task consists in recognizing the ancestors when the network is trained exclusively with their descendents. Explicit results for the dependence of the equilibrium properties of a Q=3-state model and a Q=infinity-state model are obtained in the form of phase diagrams and categorization curves. A strong improvement of the categorization ability is found when the network is trained with examples of low activity. The categorization ability is found to be robust to finite threshold and synaptic noise. The Almeida-Thouless lines that limit the validity of the replica-symmetric results, are also obtained.