On Geometric Algebra representation of Binary Spatter Codes

Distributed representation is a way of representing information in a pattern of activation over a set of neurons, in which each concept is represented by activation over multiple neurons, and each neuron participates in the representation of multiple concepts [1]. Examples of distributed representations include Recursive Auto-Associative Memory (RAAM) [2], Tensor Product Representations [3], Holographic Reduced Representations (HRRs) [4, 5], and Binary Spatter Codes (BSC) [6, 7, 8]. BSC is a powerful and simple method of representing hierarchical structures in connectionist systems and may be regarded as a binary version of HRRs. Yet, BSC has some drawbacks associated with the representation of chunking. This is why different versions of BSC can be found in the literature. In [6, 7] chunking is given by a majority-rule thresholded addition of binary strings, an operation that often discards a lot of important information. In [8] the ordinary addition is employed, and bits are parametrized differently. The main message we want to convey in this paper is that there exists a very natural representation of BSC at the level of Clifford algebras. Binding of vectors is here performed by means of the Clifford product and chunking is just ordinary addition. Since Cliford algebras possess a geometric interpretation in terms of Geometric Algebra (GA) [17, 18, 19], the cognitive structures processed in BSC or HHRs obtain a geometric content. This is philosophically consistent with many other approaches where cognition is interpreted in geometric terms [14, 22]. Of particular relevance may be the links to neural computation whose GA and HRR versions were formulated by different authors (cf. [5, 29, 30]. The present paper can be also seen in a wider context of a “quantum structures” approach to cognitive problems we have outlined elsewhere [9, 10, 11, 12, 13]. Cartan’s representation of GA in terms of tensor products of Pauli matrices introduces formal links to quantum computation (cf. [23, 24, 25, 26]). The philosophy we advocate here is also not that far from the approach of Widdows, where both geometric an “quantum” aspects play an important role [14, 15, 16]. It should be stressed that the GA calculus has already proved to be a powerful tool in applied branches of computer science (computer vision [20], robotics [21]). GA is a comprehensive language that simplified and integrated many branches of classical and quantum physics [31]. One may hope that it will play a similar role in cognitive science.