Brain-state-in-a-box neural networks with asymmetric coefficients

Vandenberghe,; Vandewalle,

doi:10.1109/ijcnn.1989.118642

Cited by 6 publications

(2 citation statements)

References 15 publications

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“…However, we should take into account that equivalent recurrent networks may be unstable. There are some special cases where stability is guaranteed [101].…”

Section: Theorem 4 ([105]) For Any Acyclic Asymmetric Neural Networkmentioning

confidence: 99%

Neural networks as systems for recognizing patterns

Smetanin¹

1998

J Math Sci

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1 This is the reason for the diversity of terms used to denote neural networks: coneural connectionist models, distributed systems, neuromorphous systems, recognition machines, associative memory. Suppose that the parameter p meets the conditions 0 < p < 1]2. The equilibrium state v is stable and possesses the attraction domain with radius pn if for any vector x such that Iv -x I < pn, I I is a network with the input vector x falling in the equilibrium position v. Here I I denotes the Hamming distance.Definition 3. Absolute stable capacity pM (n, p).For any e > 0, provided that m < p0,(n, p), the probability P{all m vectors are equilibrium states with attraction domain of radius pn} > 1 -e -o(1)}.Definition 4. Relative stable capacity pas(np).For any e > 0, provided that m < pos(n,p), the mean M{the number of vectors from the set that correspond to equilibrium states with attraction domain of radius pn} > m(1 -e -o(1)}. For nonrecurrent neural networks, the notion of equilibrium states loses its meaning, and the capacity is defined as the maximum value of input-output pairs which can be implemented with the use of the model considered. In this case, we can define the same types of capacity as for recurrent networks.Suppose that H= is a class of neural networks with n inputs from the region X,, with a single output from the region Y,, = {-1,1}. We also assume that VC-dim(Hn) = d,, and m samples of input and output signals are chosen independently in accordance with the distribution Dn within X~ x Y,~.Definition 5. The sequence pt,(n) is referred to as the lower capacity of the class H,~ if, for any arbitrarily small e > 0 and n --4 cr for any distribution D= the probability P{H,, implements all the specified relations between inputs and outputs} --* 1 provided that m < (1 -r Definition 6. The sequence pu(n) is referred to as the upper capacity of the class Hn if, for any arbitrarily small e > 0 and n --* o% for any distribution D~, the probability P{H,, implements all the specified relations between inputs and outputs} ~ 1 provided that m > (1 + e)pu(n).Definition 7. The sequence p(n) is called a capacity if it simultaneously satisfies the definitions of the lower and upper capacities.Stable capacities for nonrecurrent networks can be defined by analogy with recurrent networks. The notion of the capacity for nonrecurrent networks is closely associated with the Vapnik-Chervonenkis dimension.

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“…However, we should take into account that equivalent recurrent networks may be unstable. There are some special cases where stability is guaranteed [101].…”

Section: Theorem 4 ([105]) For Any Acyclic Asymmetric Neural Networkmentioning

confidence: 99%

Neural networks as systems for recognizing patterns

Smetanin¹

1998

J Math Sci

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“…Since the paper by Anderson and his coworkers [l], the BSB model has been investigated by many authors: Greenberg [2], Grossberg [3], Vandenberghe and Vandewalle [4], and Hui and Zak [5]. The main results of these works will not be reviewed here for sake of brevity.…”

Section: Introductionmentioning

confidence: 98%

A synthesis procedure for brain-state-in-a-box neural networks

Perfetti

1995

IEEE Trans. Neural Netw.

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In this paper, some new qualitative properties of discrete-time neural networks based on the "brain-state-in-a-box" model are presented. These properties concern both the characterization of equilibrium points and the global dynamical behavior. Next, the analysis results are used as guidelines in developing an efficient synthesis procedure for networks that function as associative memories. A constrained design algorithm is presented that gives completely stable dynamical neural networks sharing some interesting features. It is guaranteed the absence of nonbinary stable equilibria, that is stable states with nonsaturated components. It is guaranteed that in close proximity (Hamming distance one) of the stored patterns there is no other binary equilibrium point. Moreover, the presented method allows one to optimize a design parameter that controls the size of the attraction basins of the stored vectors and the accuracy needed in a digital realization of the network.

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On the stability analysis of cellular neural networks

Savacı

Vandewalle²

1993

IEEE Trans. Circuits Syst. I

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Brain-state-in-a-box neural networks with asymmetric coefficients

Cited by 6 publications

References 15 publications

Neural networks as systems for recognizing patterns

Neural networks as systems for recognizing patterns

A synthesis procedure for brain-state-in-a-box neural networks

On the stability analysis of cellular neural networks

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