Design of a universal, two-layered neural network derived from the PLI theory

The if-and-only-if (IFF) condition that a set of M analog-to-digital vector-mapping relations can be learned by a one-layered-feed-forward neural network (OLNN) is that all the input analog vectors dichotomized by the i-th output bit must be positively, linearly independent, or PLI. If they are not PLI, then the OLNN just cannot learn no matter what learning rules is employed because the solution of the connection matrix does not exist mathematically. However, in this case, one can still design a parallel-cascaded, two-layered, perceptron (PCTLP) to achieve this general mapping goal. The design principle of this "universal" neural network is derived from the major mathematical properties of the PLI theory - changing the output bits of the dependent relations existing among the dichotomized input vectors to make the PLD relations PLI. Then with a vector concatenation technique, the required mapping can still be learned by this PCTLP system with very high efficiency. This paper will report in detail the mathematical derivation of the, general design principle and the design procedures of the PCTLP neural network system. It then will be verified in general by a practical numerical example.