Retrieval and chaos in extremely dilutedQ-Ising neural networks

Bollé, Désiré; Shim, Gyoung Moo; Vinck, B; Zagrebnov, Valentin A.

doi:10.1007/bf02188571

Cited by 30 publications

(41 citation statements)

References 17 publications

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“…4, where we omit again the thermodynamic transitions and the long-dashed lines indicate now the onset of the binary-network behavior. Note that the disappearance of the ordered phase takes place at θ = 1/2 for any finite connectivity, as in the case of the fully connected network and in networks of different architecture, like the extremely asymmetric diluted and the Q-Ising layered network [15,13,14]. In the case of the extremely diluted networks, the retrieval phase boundaries have a reentrance for θ ≥ 1 2 [19].…”

Section: The Continuous Response Networkmentioning

confidence: 96%

Retrieval behavior and thermodynamic properties of symmetrically dilutedQ-Ising neural networks

Erichsen²

2001

Phys. Rev. E

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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 = ∞ 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.

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Section: The Continuous Response Networkmentioning

confidence: 96%

Retrieval behavior and thermodynamic properties of symmetrically dilutedQ-Ising neural networks

Erichsen²

2001

Phys. Rev. E

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“…In the extremely diluted and layered Q-Ising models the evolution equations for the order parameters do not change their form as time progresses, such that the fixed-point equations are obtained immediately by leaving out the time dependence (see [2], [4]). This still allows small fluctuations in the configurations {σ i }.…”

Section: Fixed-point Equationsmentioning

confidence: 99%

“…If the dilution is asymmetric [2], even all feedback disappears and the local field is simply Gaussian distributed. For the special case of Q = 2 these results agree with the results available for the first three time steps in the literature [5]- [6].…”

Section: A3: Third Step Dynamicsmentioning

confidence: 99%

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Bollé

Jongen

Shim

1999

Journal of Statistical Physics

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The parallel dynamics of extremely diluted symmetric Q-Ising neural networks is studied for arbitrary Q using a probabilistic approach. In spite of the extremely diluted architecture the feedback correlations arising from the symmetry prevent a closed-form solution in contrast with the extremely diluted asymmetric model. A recursive scheme is found determining the complete time evolution of the order parameters taking into account all feedback. It is based upon the evolution of the distribution of the local field, as in the fully connected model. As an illustrative example an explicit analysis is carried out for the Q = 2 and Q = 3 model. These results agree with and extend the partial results existing for Q = 2. For Q > 2 the analysis is entirely new. Finally, equilibrium fixed-point equations are derived and a capacity-gain function diagram is obtained.

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“…The dynamics of this model is then studied following standard methods involving a signal-to-noise analysis (see, e.g., [19], [20], [21]). At this point we recall that it is justified to first dilute the system by taking the limit N → ∞ and second, in the diluted system, to apply the law of large numbers (LLN) and the central limit theorem (CLT) by taking the limit C → ∞.…”

Section: The Diluted Dynamicsmentioning

confidence: 99%

Categorization by a three-state attractor neural network

Domínguez

Bollé

1997

Phys. Rev. E

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The categorization properties of an attractor network of threestate neurons which infers three-state concepts from examples are studied. The evolution equations governing the parallel dynamics at zero temperature for the overlap between the state of the network and the examples, the state of the network and the concepts as well as the neuron activity are discussed in the limit of extreme dilution. A transition from a retrieval region to a categorization region is found when the number of examples or their correlations are increased. If the pattern activity is small enough, the examples (concepts) are very well retrieved (categorized) for an appropriate choice of the zero-activity threshold of the neurons.

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Retrieval and chaos in extremely dilutedQ-Ising neural networks

Cited by 30 publications

References 17 publications

Retrieval behavior and thermodynamic properties of symmetrically dilutedQ-Ising neural networks

Retrieval behavior and thermodynamic properties of symmetrically dilutedQ-Ising neural networks

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Categorization by a three-state attractor neural network

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