Two or three things that we (intend to) know about Hopfield and Tank networks

This work aims at reviewing some of the main issues that are under research in the field of Hopfield networks. In particular, the feasibility of the Hopfield network as a practical optimization method is addressed. Together with the current results, the main directions that de- serve ongoing analysis are shown. Besides, some suggestions are provided in order to identify lines that are at an impasse point, where there is no evidence that further research will be fruitful, or topics that nowadays can just be considered as historically interesting.

[1]  Francisco Sandoval Hernández,et al.  Parametric identification of robotic systems with stable time-varying Hopfield networks , 2004, Neural Computing & Applications.

[2]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[3]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

[4]  Francisco Sandoval Hernández,et al.  Associating arbitrary-order energy functions to an artificial neural network: Implications concerning the resolution of optimization problems , 1997, Neurocomputing.

[5]  Francisco Sandoval,et al.  Combinatorial Optimization via Discrete Gradient Implementation of Hopfield Neural Networks , 2003 .

[6]  Alexander S. Poznyak,et al.  Differential Neural Networks for Robust Nonlinear Control , 2004, IEEE Transactions on Neural Networks.

[7]  Gonzalo Joya,et al.  Numerical implementation of continuous Hop eld networks for optimization , 2001 .

[8]  Infinite dimensional Hopfield neural networks , 2001 .

[9]  Francisco Sandoval Hernández,et al.  Spurious Minima and Basins of Attraction in Higher-Order Hopfield Networks , 2003, IWANN.

[10]  Jochen J. Steil.  Local stability of recurrent networks with time-varying weights and inputs , 2002, Neurocomputing.

[11]  Kwok-Wo Wong,et al.  Criteria for exponential stability of Cohen-Grossberg neural networks , 2004, Neural Networks.

[12]  Jinde Cao,et al.  Estimation on Domain of Attraction and Convergence Rate of Hopfield Continuous Feedback Neural Networks , 2001, J. Comput. Syst. Sci..

[13]  Francisco Sandoval Hernández,et al.  Hopfield Neural Networks for Parametric Identification of Dynamical Systems , 2004, Neural Processing Letters.

[14]  Youichi Kobuchi.  State evaluation functions and Lyapunov functions for neural networks , 1991, Neural Networks.

[15]  T. Kohonen.  Analysis of a simple self-organizing process , 1982, Biological Cybernetics.

[16]  Kate Smith-Miles.  An argument for abandoning the travelling salesman problem as a neural-network benchmark , 1996, IEEE Trans. Neural Networks.

[17]  Gonzalo Joya,et al.  Application of high-order hopfield neural networks to the solution of diophantine equations , 1991 .

[18]  K Smith,et al.  An argument for abandoning the travelling salesman problem as a neural-network benchmark. , 1996, IEEE transactions on neural networks.

[19]  Mathukumalli Vidyasagar.  Minimum-seeking properties of analog neural networks with multilinear objective functions , 1995, IEEE Trans. Autom. Control..

[20]  Francisco Sandoval Hernández,et al.  Hopfield neural networks for optimization: study of the different dynamics , 2002, Neurocomputing.

[21]  Roberto P. J. Perazzo,et al.  Associative Memories in Infinite Dimensional Spaces , 2000, Neural Processing Letters.

[22]  Francisco Sandoval Hernández,et al.  Dynamical Analysis of Continuous Higher-Order Hopfield Networks for Combinatorial Optimization , 2005, Neural Computation.

[23]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[24]  X. Mao,et al.  Stochastic Hopfield neural networks , 2003 .

[25]  Teuvo Kohonen,et al.  Self-organized formation of topologically correct feature maps , 2004, Biological Cybernetics.

[26]  Gang Feng,et al.  On the Convergence and Parameter Relation of Discrete-Time Continuous-State Hopfield Networks with Self-Interaction Neurons , 2001 .

[27]  Yoshikane Takahashi,et al.  A neural network theory for constrained optimization , 1999, Neurocomputing.

[28]  Francisco Sandoval Hernández,et al.  Continuous-State Hopfield Dynamics Based on Implicit Numerical Methods , 2002, ICANN.

[29]  Kate Smith-Miles,et al.  Neural Networks for Combinatorial Optimization: A Review of More Than a Decade of Research , 1999, INFORMS J. Comput..

[30]  K. Gopalsamy,et al.  Dynamics of a class of discete-time neural networks and their comtinuous-time counterparts , 2000 .

[31]  G. Pawley,et al.  On the stability of the Travelling Salesman Problem algorithm of Hopfield and Tank , 2004, Biological Cybernetics.

[32]  Stephen Grossberg,et al.  Absolute stability of global pattern formation and parallel memory storage by competitive neural networks , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[33]  Mathukumalli Vidyasagar.  Are analog neural networks better than binary neural networks? , 1998 .

[34]  Xin Wang,et al.  Absence of Cycles in Symmetric Neural Networks , 1998, Neural Computation.

[35]  J. J. Hopfield,et al.  “Neural” computation of decisions in optimization problems , 1985, Biological Cybernetics.

[36]  Shun-ichi Amari,et al.  New theorems on global convergence of some dynamical systems , 2001, Neural Networks.

[37]  Jihan Zhu,et al.  FPGA Implementations of Neural Networks - A Survey of a Decade of Progress , 2003, FPL.

[38]  N. J. Cohen,et al.  Higher-Order Boltzmann Machines , 1986 .

[39]  Gilles Pagès,et al.  Two or three things that we know about the Kohonen algorithm , 1994, ESANN.

[40]  Tariq Samad,et al.  High-order Hopfield and Tank optimization networks , 1990, Parallel Comput..

[41]  M. V. Velzen,et al.  Self-organizing maps , 2007 .

[42]  Peter Tiño,et al.  Attractive Periodic Sets in Discrete-Time Recurrent Networks (with Emphasis on Fixed-Point Stability and Bifurcations in Two-Neuron Networks) , 2001, Neural Computation.

[43]  Satoshi Matsuda.  “Optimal” neural representation of higher order for traveling salesman problems , 2002 .

[44]  S. Abe.  Theories on the Hopfield neural networks , 1989, International 1989 Joint Conference on Neural Networks.

[45]  Yoshiyasu Takefuji,et al.  Artificial neural networks for four-coloring map problems and K-colorability problems , 1991 .

[46]  Francisco Sandoval Hernández,et al.  Numerical implementation of continuous Hopfield networks for optimization , 2001, ESANN.

[47]  A. Bountis.  Dynamical Systems And Numerical Analysis , 1997, IEEE Computational Science and Engineering.

[48]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.