Implementing invariances in high order neural nets

In this paper we examine the properties of high order neuron-like adaptive learning units whose output is invariant under an arbitrary finite group of transformations on the input space. The transformation invariance is imposed by averaging the input of each unit over a transformation group, thus eliminating the capacity of the units to detect features which are incompatible with the imposed group invariance. This averaging process also generates equivalence classes of interactions among the units, and thus allows a collapse of the interaction weight matrix, reducing the number of high order terms. As an example, we discuss the implementation of two types of translation invariance.