Absolute Exponential Stability of Recurrent Neural Networks With Generalized Activation Function
In this paper, the recurrent neural networks (RNNs) with a generalized activation function class is proposed. In this proposed model, every component of the neuron's activation function belongs to a convex hull which is bounded by two odd symmetric piecewise linear functions that are convex or...
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| Published in | IEEE transactions on neural networks Vol. 19; no. 6; pp. 1075 - 1089 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
New York, NY
IEEE
01.06.2008
Institute of Electrical and Electronics Engineers |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1045-9227 1941-0093 |
| DOI | 10.1109/TNN.2007.2000060 |
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| Abstract | In this paper, the recurrent neural networks (RNNs) with a generalized activation function class is proposed. In this proposed model, every component of the neuron's activation function belongs to a convex hull which is bounded by two odd symmetric piecewise linear functions that are convex or concave over the real space. All of the convex hulls are composed of generalized activation function classes. The novel activation function class is not only with a more flexible and more specific description of the activation functions than other function classes but it also generalizes some traditional activation function classes. The absolute exponential stability (AEST) of the RNN with a generalized activation function class is studied through three steps. The first step is to demonstrate the global exponential stability (GES) of the equilibrium point of original RNN with a generalized activation function being equivalent to that of RNN under all vertex functions of convex hull. The second step transforms the RNN under every vertex activation function into neural networks under an array of saturated linear activation functions. Because the GES of the equilibrium point of three systems are equivalent, the next stability analysis focuses on the GES of the equilibrium point of RNN system under an array of saturated linear activation functions. The last step is to study both the existence of equilibrium point and the GES of the RNN under saturated linear activation functions using the theory of M -matrix. In the end, a two-neuron RNN with a generalized activation function is constructed to show the effectiveness of our results. |
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| AbstractList | In this paper, the recurrent neural networks (RNNs) with a generalized activation function class is proposed. In this proposed model, every component of the neuron's activation function belongs to a convex hull which is bounded by two odd symmetric piecewise linear functions that are convex or concave over the real space. All of the convex hulls are composed of generalized activation function classes. The novel activation function class is not only with a more flexible and more specific description of the activation functions than other function classes but it also generalizes some traditional activation function classes. The absolute exponential stability (AEST) of the RNN with a generalized activation function class is studied through three steps. The first step is to demonstrate the global exponential stability (GES) of the equilibrium point of original RNN with a generalized activation function being equivalent to that of RNN under all vertex functions of convex hull. The second step transforms the RNN under every vertex activation function into neural networks under an array of saturated linear activation functions. Because the GES of the equilibrium point of three systems are equivalent, the next stability analysis focuses on the GES of the equilibrium point of RNN system under an array of saturated linear activation functions. The last step is to study both the existence of equilibrium point and the GES of the RNN under saturated linear activation functions using the theory of M-matrix. In the end, a two-neuron RNN with a generalized activation function is constructed to show the effectiveness of our results. In this paper, the recurrent neural networks (RNNs) with a generalized activation function class is proposed. In this proposed model, every component of the neuron's activation function belongs to a convex hull which is bounded by two odd symmetric piecewise linear functions that are convex or concave over the real space. All of the convex hulls are composed of generalized activation function classes. The novel activation function class is not only with a more flexible and more specific description of the activation functions than other function classes but it also generalizes some traditional activation function classes. The absolute exponential stability (AEST) of the RNN with a generalized activation function class is studied through three steps. The first step is to demonstrate the global exponential stability (GES) of the equilibrium point of original RNN with a generalized activation function being equivalent to that of RNN under all vertex functions of convex hull. The second step transforms the RNN under every vertex activation function into neural networks under an array of saturated linear activation functions. Because the GES of the equilibrium point of three systems are equivalent, the next stability analysis focuses on the GES of the equilibrium point of RNN system under an array of saturated linear activation functions. The last step is to study both the existence of equilibrium point and the GES of the RNN under saturated linear activation functions using the theory of M-matrix. In the end, a two-neuron RNN with a generalized activation function is constructed to show the effectiveness of our results.In this paper, the recurrent neural networks (RNNs) with a generalized activation function class is proposed. In this proposed model, every component of the neuron's activation function belongs to a convex hull which is bounded by two odd symmetric piecewise linear functions that are convex or concave over the real space. All of the convex hulls are composed of generalized activation function classes. The novel activation function class is not only with a more flexible and more specific description of the activation functions than other function classes but it also generalizes some traditional activation function classes. The absolute exponential stability (AEST) of the RNN with a generalized activation function class is studied through three steps. The first step is to demonstrate the global exponential stability (GES) of the equilibrium point of original RNN with a generalized activation function being equivalent to that of RNN under all vertex functions of convex hull. The second step transforms the RNN under every vertex activation function into neural networks under an array of saturated linear activation functions. Because the GES of the equilibrium point of three systems are equivalent, the next stability analysis focuses on the GES of the equilibrium point of RNN system under an array of saturated linear activation functions. The last step is to study both the existence of equilibrium point and the GES of the RNN under saturated linear activation functions using the theory of M-matrix. In the end, a two-neuron RNN with a generalized activation function is constructed to show the effectiveness of our results. |
| Author | Youxian Sun Jun Xu Jinshan Tang Yong-Yan Cao |
| Author_xml | – sequence: 1 givenname: Jun surname: Xu fullname: Xu, Jun email: xujung@gmail.com organization: System Research Institute & Department of Advanced Technologies, Alcorn State University, Alcorn State, MS 39096 USA. xujung@gmail.com – sequence: 2 givenname: Yong-Yan surname: Cao fullname: Cao, Yong-Yan – sequence: 3 givenname: Youxian surname: Sun fullname: Sun, Youxian – sequence: 4 givenname: Jinshan surname: Tang fullname: Tang, Jinshan |
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| Keywords | Recurrent neural nets recurrent neural networks (RNNs) Generalized function Absolute exponential stability (AEST) Symmetric function Neural network Exponential stability Modeling Absolute stability Equilibrium point Convex hull piecewise linear function Activation function piecewise function Convex function generalized activation function class Piecewise linearization |
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| SubjectTerms | Absolute exponential stability (AEST) Activation Activation analysis Animals Applied sciences Arrays Artificial Intelligence Asymptotic stability Computer science; control theory; systems Computer Simulation Connectionism. Neural networks Convergence convex hull Exact sciences and technology generalized activation function class Hulls Hulls (structures) Industrial control Information technology Laboratories Mathematical models Models, Neurological Neural networks Neural Networks (Computer) Neurons - physiology Nonlinear Dynamics Pattern Recognition, Automated - methods piecewise linear function Piecewise linear techniques Recurrent neural networks recurrent neural networks (RNNs) Stability Stability analysis Sun Time Factors |
| Title | Absolute Exponential Stability of Recurrent Neural Networks With Generalized Activation Function |
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