Computation with Cliiord Valued Feed-forward Networks

Recent research has focused on feed-forward networks with complex weights and activation values such as GK92, Hir92b, Hir92a, Hir93]. This paper extends this formalism to feed-forward networks with weight and activation values taken from a Cliiord algebra (see also PB92, PB94b]). A Cliiord algebra is a multi-dimensional generalization of the complex numbers and the Quaternions. Essentially a Cliiord algebra is obtained by extending vector spaces to allow an associative multiplication compatible with the natural metric on the vector space. This paper presents an extension of the well known back-error propagation algorithm to Cliiord valued feed-forward networks, and presents some experimental results with simple encoder-decoder problems. A discussion of the diierence between real and Cliiord valued networks is also included. Finally a Universal Approximation similar to the results found in HSW89] is proved.