Geometric analogue of holographic reduced representation

Diederik Aerts; Marek Czachor; Moor Bart De

doi:10.1016/j.jmp.2009.02.005

Geometric analogue of holographic reduced representation

Abstract

Holographic reduced representations (HRRs) are distributed representations of cognitive structuresbased on superpositions of convolution-bound n-tuples. Restricting HRRs to n-tuples consisting of 1,one reinterprets the variable binding as a representation of the additive group of binary n-tupleswith addition modulo 2. Since convolutions are not defined for vectors, the HRRs cannot be directlyassociated with geometric structures. Geometric analogues of HRRs are obtained if one considers aprojective representation of the same group in the space of blades (geometric products of basis vectors)associated with an arbitrary n-dimensional Euclidean (or pseudo-Euclidean) space. Switching to matrixrepresentations of Clifford algebras, one can always turn a geometric analogue of an HRR into a form ofmatrix distributed representation. In typical applications the resulting matrices are sparse, so that thematrix representation is less efficient than the representation directly employing the rules of geometricalgebra. A yet more efficient procedure is based on `projected products', a hierarchy of geometricallymeaningful n-tuple multiplication rules obtained by combining geometric products with projectionson relevant multivector subspaces. In terms of dimensionality the geometric analogues of HRRs are inbetween holographic and tensor-product representations.

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DOI:: Digital Object Identifier (open in new tab) 10.1016/j.jmp.2009.02.005
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Keywords

GEOMETRIC ANALOGUE

Details

Category:: Articles
Type:: artykuł w czasopiśmie wyróżnionym w JCR
Published in:: Journal of Mathematical Psychology no. 53, pages 389 - 398,
ISSN: 0022-2496
Language:: English
Publication year:: 2009
Bibliographic description:: Aerts D., Czachor M., De M.: Geometric analogue of holographic reduced representation// Journal of Mathematical Psychology. -Vol. 53, nr. Iss 5, October. (2009), s.389-398
DOI:: Digital Object Identifier (open in new tab) 10.1016/j.jmp.2009.02.005
Verified by:: Gdańsk University of Technology