Properties of Neural Networks Studied from ND Geometry and Discrete Algebra

C.-L.J. Hu (USA)


Feedback and Feed-forward neural networks, N dimension geometry.


For a one-layered neural network, NN, containing discrete sign-function neurons, (either a feedback NN, FBNN, or a feed-forward NN, FFNN,) the nonlinear properties of this network can be studied very efficiently using simple discrete mathematics. This paper summarizes the discrete-formulation of the control matrix equation of the problem. Then two methods for solving this matrix equation are used. One is for the FFNN, in which we reduced the control equation to a set of linear strict inequalities. The other one is for FBNN, in which we use simple finite iterative method of solving this nonlinear matrix equation. The derivation of the major anomalous properties of both systems are then obtained. These anomalous properties include optimum robustness, noise-controlled digital recalls, domain of attractions in the state space, eigen-state storage, associative storage, content-addressable recall, fault tolerant recall, capacity of storage, binary oscillating states, limit-cycles in the state space, etc.. The physical origin and the systematic trend of the derivation of these properties are easily seen from the state-space geometrical analysis.

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