Analog Perceptron: Its Decomposition and Order

A theory of computational geometry, in which it is discussed how geometrical properties of pattern are reflected in the structure of “parallel” information processing machine like Perceptron, was proposed by Minsky and Papert (1969) . The theory has been extended by Uesaka (1971) to analog Perceptrons with real-valued inputs and output. One of the concepts playing a central role in this theory is “order” which expresses a certain kind of complexity of a parallel-type machine. Thus, the main theme of the theory is to establish the methodology for determining the order of a given Perceptron. Although the group-invariance theorem, the classification theorem, etc., were already obtained as methods for evaluating the order, their applications to a Perceptron demand that it must have a certain kind of property—e.g., it must be a symmetrical function. Thus, in the present paper, new methods applicable to the wider range of Perceptrons will be given. In those methods, a given Perceptron is first decomposed in additive and/or multiplicative form, and next the order of the Perceptron is determined from the orders of decomposed Perceptrons. Those methods enable us to evaluate the order of a Perceptron expressible with a polynominal or a real analytic function.