75 research outputs found
A layered neural network with three-state neurons optimizing the mutual information
The time evolution of an exactly solvable layered feedforward neural network
with three-state neurons and optimizing the mutual information is studied for
arbitrary synaptic noise (temperature). Detailed stationary
temperature-capacity and capacity-activity phase diagrams are obtained. The
model exhibits pattern retrieval, pattern-fluctuation retrieval and spin-glass
phases. It is found that there is an improved performance in the form of both a
larger critical capacity and information content compared with three-state
Ising-type layered network models. Flow diagrams reveal that saddle-point
solutions associated with fluctuation overlaps slow down considerably the flow
of the network states towards the stable fixed-points.Comment: 17 pages Latex including 6 eps-figure
Synchronous versus sequential updating in the three-state Ising neural network with variable dilution
The three-state Ising neural network with synchronous updating and variable
dilution is discussed starting from the appropriate Hamiltonians. The
thermodynamic and retrieval properties are examined using replica mean-field
theory. Capacity-temperature phase diagrams are derived for several values of
the pattern activity and different gradations of dilution, and the information
content is calculated. The results are compared with those for sequential
updating. The effect of self-coupling is established. Also the dynamics is
studied using the generating function technique for both synchronous and
sequential updating. Typical flow diagrams for the overlap order parameter are
presented. The differences with the signal-to-noise approach are outlined.Comment: 21 pages Latex, 12 eps figures and 1 ps figur
Correlated patterns in non-monotonic graded-response perceptrons
The optimal capacity of graded-response perceptrons storing biased and
spatially correlated patterns with non-monotonic input-output relations is
studied. It is shown that only the structure of the output patterns is
important for the overall performance of the perceptrons.Comment: 4 pages, 4 figure
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 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.Comment: 21 pages, revte
Thresholds in layered neural networks with variable activity
The inclusion of a threshold in the dynamics of layered neural networks with
variable activity is studied at arbitrary temperature. In particular, the
effects on the retrieval quality of a self-controlled threshold obtained by
forcing the neural activity to stay equal to the activity of the stored paterns
during the whole retrieval process, are compared with those of a threshold
chosen externally for every loading and every temperature through optimisation
of the mutual information content of the network. Numerical results, mostly
concerning low activity networks are discussed.Comment: 15 pages, Latex2e, 6 eps figure
Statics and dynamics of an Ashkin-Teller neural network with low loading
An Ashkin-Teller neural network, allowing for two types of neurons is
considered in the case of low loading as a function of the strength of the
respective couplings between these neurons. The storage and retrieval of
embedded patterns built from the two types of neurons, with different degrees
of (in)dependence is studied. In particular, thermodynamic properties including
the existence and stability of Mattis states are discussed. Furthermore, the
dynamic behaviour is examined by deriving flow equations for the macroscopic
overlap. It is found that for linked patterns the model shows better retrieval
properties than a corresponding Hopfield model.Comment: 20 pages, 6 figures, Latex with postscript figures in one tar.gz fil
Thermodynamic properties of extremely diluted symmetric Q-Ising neural networks
Using the replica-symmetric mean-field theory approach the thermodynamic and
retrieval properties of extremely diluted {\it symmetric} -Ising neural
networks are studied. In particular, capacity-gain parameter and
capacity-temperature phase diagrams are derived for and .
The zero-temperature results are compared with those obtained from a study of
the dynamics of the model. Furthermore, the de Almeida-Thouless line is
determined. Where appropriate, the difference with other -Ising
architectures is outlined.Comment: 16 pages Latex including 6 eps-figures. Corrections, also in most of
the figures have been mad
A canonical ensemble approach to graded-response perceptrons
Perceptrons with graded input-output relations and a limited output precision
are studied within the Gardner-Derrida canonical ensemble approach. Soft non-
negative error measures are introduced allowing for extended retrieval
properties. In particular, the performance of these systems for a linear and
quadratic error measure, corresponding to the perceptron respectively the
adaline learning algorithm, is compared with the performance for a rigid error
measure, simply counting the number of errors. Replica-symmetry-breaking
effects are evaluated.Comment: 26 pages, 10 ps figure
The signal-to-noise analysis of the Little-Hopfield model revisited
Using the generating functional analysis an exact recursion relation is
derived for the time evolution of the effective local field of the fully
connected Little-Hopfield model. It is shown that, by leaving out the feedback
correlations arising from earlier times in this effective dynamics, one
precisely finds the recursion relations usually employed in the signal-to-noise
approach. The consequences of this approximation as well as the physics behind
it are discussed. In particular, it is pointed out why it is hard to notice the
effects, especially for model parameters corresponding to retrieval. Numerical
simulations confirm these findings. The signal-to-noise analysis is then
extended to include all correlations, making it a full theory for dynamics at
the level of the generating functional analysis. The results are applied to the
frequently employed extremely diluted (a)symmetric architectures and to
sequence processing networks.Comment: 26 pages, 3 figure
Small-world hypergraphs on a bond-disordered Bethe lattice
We study the thermodynamic properties of spin systems with bond-disorder on
small-world hypergraphs, obtained by superimposing a one-dimensional Ising
chain onto a random Bethe graph with p-spin interactions. Using transfer-matrix
techniques, we derive fixed-point equations describing the relevant order
parameters and the free energy, both in the replica symmetric and one step
replica symmetry breaking approximation. We determine the static and dynamic
ferromagnetic transition and the spinglass transition within replica symmetry
for all temperatures, and demonstrate corrections to these results when one
step replica symmetry breaking is taken into account. The results obtained are
in agreement with Monte-Carlo simulations.Comment: 9 pages, 4 figure
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