# Geometric Algebra Model Of Distributed Representations

Agnieszka Patyk Abstract Formalism based on GA is an alternative to distributed representation models developed so far - Smolensky's tensor product, Holographic Reduced Representations (HRR) and Binary Spatter Code (BSC). Convolutions are replaced by geometric products, interpretable in terms of geometry which seems to be the most natural language for visualization of higher concepts. This paper recalls the main ideas behind the GA model and investigates recognition test results using both inner product and a clipped version of matrix representation. The influence of accidental blade equality on recognition is also studied. Finally, the efficiency of the GA model is compared to that of previously developed models.

## 1 Introduction

Since the early 1980s a new idea of representing knowledge has emerged by the name of distributed representation. It has been the answer to the problems of recognition, reasoning and language processing - people accomplish these everyday tasks effortlessly, often with only noisy and partial information, while computational resources required for these assignments are enormous. To this day many models have been built, in which arbitrary variable bindings, short sequences of various lengths and predicates are all usually represented as fixed-width high dimensional vectors that encode information throughout the elements. In 1990 Smolensky [14]
described how tensor product algebra provides a framework for the distributed representation of recursive structures. Unfortunately, Smolensky's tensor product does not meet all criteria of reduced representations as the size of the tensor increases with the size of the structure. Nevertheless, Smolensky and Dolan [15] have shown, 1