Analog Perceptrons: On Additive Representation of Functions

The theory of computational geometry in Perceptrons ( Rosenblatt, 1962 ), developed by M. Minsky and S. Papert (1969) , is extended to “Analog Perceptrons” with real-valued input and out-put. Mathematically, our problem is to determine the order of a function, i.e., the smallest number of variables necessary to make an additive representation of the function by employing partial functions of the smaller number of variables. Mathematical tools, called the group-invariance theorem, the classification theorem and the collapsing theorem, are given which are useful for evaluating the order of analog Perceptrons. These are also applied for several analog Perceptrons.