Differential Perceptrons

Abstract

As originally proposed, perceptrons were machines that scanned a discrete retina and combined the data gathered in a linear fashion to make decisions about the figure presented on the retina. This paper considers differential perceptions, which view a continuous retina. Thus, instead of summing the results of predicates, we must now integrate. This involves setting up a predicate space which transforms the typical perceptron sum, Ea(p)a(f), into Esacp,f(p)dp, where f is the figure on the retina, i.e. in the differential case, the figure is viewed as a function on the predicate space. We show that differential perceptrons are equivalent to perceptrons on the class of figures that fit exactly onto a sufficiently small square grid. By investigating predicates of various geometric transformations, we discover that translation and symmetry can be computed in finite order using finite coefficients in both continuous and discrete cases. We also note that in the perceptron scheme, combining data linearly implies the ability to combine data in a polynomial fashion.