A comparison of geometric analogues of holographic reduced representations, original holographic reduced representations and binary spatter codes

Geometric Analogues of Holographic Reduced Representations (GA HRR) employ role-filler binding based on geometric products. Atomic objects are real-valued vectors in n-dimensional Euclidean space and complex statements belong to a hierarchy of multivectors. The paper reports a battery of tests aimed at comparison of GA HRR with Holographic Reduced Representation (HRR) and Binary Spatter Codes (BSC). Firstly, we perform a test of GA HRR which is analogous to the one proposed by Plate in [13]. Plate's simulation involved several thousand 512-dimensional vectors stored in clean-up memory. The purpose was to study efficiency of HRR but also to provide a counterexample to claims that role-filler representations do not permit one component of a relation to be retrieved given the others. We repeat Plate's test on a continuous version of GA HRR — GA
c
(as opposed to its discrete version described in [12]) and compare the results with the original HRR and BSC. The object of the test is to construct statements concerning multiplication and addition. For example, “2·3 = 6” is constructed as times
2,3
= times+operand∗(num
2
+ num
3
)+result∗num
6
. To look up this vector one then constructs a similar statement with one of the components missing and checks whether it points correctly to times
2,3
. We concentrate on comparison of recognition percentage for the three models for comparable data size, rather than on the time taken to achieve high percentage. Results show that the best models for storing and recognizing multiple similar statements are GA
c
and Binary Spatter Codes with recognition percentage highly above 90.