Analysis of nonseparable property of multi-valued multi-threshold neuron

We consider the multi-valued discrete real training set that can not be separated by one multi-valued multi-threshold neuron. Such training set is defined as linearly nonseparable set in this paper. Our objective is to use multi-valued multi-threshold neural networks to learn nonseparable training sets. First we give the method that how to determine a training set is separable or nonseparable (i.e., the necessary and sufficient condition for linearly nonseparable is given). Then we analyze the structures within linearly nonseparable sets: not all the vectors in a linearly nonseparable set are responsible for nonseparability. So the vectors in such set can be partitioned to separable vectors and nonseparable vectors. Finally, we discuss the learning problems for a linearly nonseparable set. Such set can be learned by a three-layer feedforward neural network with one hidden layer. An example throughout the paper further clarifies the results of this paper.