Geometric Analogue of Holographic Reduced Representation

TL;DR

This paper introduces a geometric analogue of holographic reduced representations (HRR), enabling basis-independent, interpretable vector operations with exact inverses, bridging HRR with geometric and quantum computation concepts.

Contribution

The paper develops a geometric version of HRR that is basis-independent and allows for exact inverse computations, unlike traditional HRR.

Findings

Geometric analogue of HRR is basis-independent.

Variable binding via geometric product allows exact inverses.

Connects reduced representations with geometric and quantum computation.

Abstract

Holographic reduced representations (HRR) are based on superpositions of convolution-bound $n$-tuples, but the $n$-tuples cannot be regarded as vectors since the formalism is basis dependent. This is why HRR cannot be associated with geometric structures. Replacing convolutions by geometric products one arrives at reduced representations analogous to HRR but interpretable in terms of geometry. Variable bindings occurring in both HRR and its geometric analogue mathematically correspond to two different representations of $Z_2\times...\times Z_2$ (the additive group of binary $n$-tuples with addition modulo 2). As opposed to standard HRR, variable binding performed by means of geometric product allows for computing exact inverses of all nonzero vectors, a procedure even simpler than approximate inverses employed in HRR. The formal structure of the new reduced representation is analogous to cartoon computation, a geometric analogue of quantum computation.