Stabilizing relaxed nonlinear FMA yields a (combinatorial) optimizer
ICONIP'12 Proceedings of the 19th international conference on Neural Information Processing - Volume Part I
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A Hopfield-like neural network, called SALU-SIR,whose system weight matrix is symmetric is presented withits mathematical analysis in [7]. However, what happens if the system matrix is unsymmetric? Is the system still stable in the unsymmetric case? In this paper, we address these importantquestions, whose answer is paramount especially when the system is to be implemented in practice.The underlying linear system of the proposed network is x(k+1) = Ax(k)+b where A is any real square unsymmetric matrix with linearly independent eigenvectors whose largest eigenvalue is real and its norm is larger than 1, and vector b is constant. Our investigations in this paper show that i) the unsymmetric case is also stable; ii) the unsymmetric case yields state-specific ultimate SIRs as compared to the system-specific ultimate SIR in the symmetric case [7], which allows us to design more complex systems. iii) the ultimate “SIR”s in the investigated unsymmetric matrix A case are equal to aiiλmax−aii , i = 1, 2, . . . , N, where Nis the number of states, aii is the diagonal elements of matrix A, and λmax is the (single or multiple) eigenvalue with maximum norm.Possible applications include binary associative memory systems, image restoration, etc in the area of artificial intelligence and cognition.