Retrieval and chaos in layeredQ-Ising neural networks

Bollé, Désiré; Shim, Gyoung Moo; Vinck, B

doi:10.1007/bf02188572

Cited by 18 publications

(23 citation statements)

References 14 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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“…Using a probabilistic approach (see, e.g., [4], [8]) we calculate the distribution of the local field for a general time step for Q ≥ 2 systems analogously to the fully connected case studied very recently [9]. This allows us to obtain recursion relations determining the full time evolution of the relevant order parameters.…”

Section: Recursive Dynamical Schemementioning

confidence: 99%

“…This implies that r µ (t+1) is Gaussian with variance a(t+1)/A, in contrast with both the fully connected architecture [9] and the layered architecture [4], where the variance contains extra terms. This finishes the treatment of the residual overlap.…”

Section: Recursive Dynamical Schemementioning

confidence: 99%

“…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%

“…[1]- [4] and the references cited therein). This has been possible because in these types of networks one knows that there are no feedback correlations as time progresses.…”

Section: Introductionmentioning

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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A layered neural network with three-state neurons optimizing the mutual information

Bollé¹,

Erichsen²,

Theumann³

2004

Physica A: Statistical Mechanics and its Applications

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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.

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Retrieval and chaos in layeredQ-Ising neural networks

Cited by 18 publications

References 14 publications

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

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

Untitled

A layered neural network with three-state neurons optimizing the mutual information

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