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Geometric Algebra Model of Distributed Representations

  • Agnieszka PatykEmail author
Chapter

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.

Keywords

Correct Answer Geometric Algebra Potential Answer Geometric Product Object Construction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Aerts, D., Czachor, M., De Moor, B.: On geometric-algebra representation of binary spatter codes. Preprint (2006). arXiv:cs/0610075 [cs.AI]
  2. 2.
    Aerts, D., Czachor, M.: Cartoon computation: Quantum-like algorithms without quantum mechanics. J. Phys. A 40, F259 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aerts, D., Czachor, M.: Tensor-product vs. geometric-product coding. Phys. Rev. A 77, 012316 (2008). arXiv:0709.1268 [quant-ph] CrossRefMathSciNetGoogle Scholar
  4. 4.
    Aerts, D., Czachor, M., De Moor, B.: Geometric analogue of holographic reduced representation. Preprint (2007). arXiv:0710.2611
  5. 5.
    Bayro-Corrochano, E.: Handbook of Geometric Computing. Springer, Berlin (2005) Google Scholar
  6. 6.
    Czachor, M.: Elementary gates for cartoon computation. J. Phys. A 40, F753 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science. Morgan-Kauffman, San Mateo (2007) Google Scholar
  8. 8.
    Hestenes, D.: Space-Time Algebra. Gordon and Breach, New York (1966) zbMATHGoogle Scholar
  9. 9.
    Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Reidel, Dordrecht (1984) zbMATHGoogle Scholar
  10. 10.
    Kanerva, P.: Binary spatter codes of ordered k-tuples. In: von der Malsburg, C., et al. (eds.) Artificial Neural Networks ICANN Proceedings. Lecture Notes in Computer Science, vol. 1112, pp. 869–873. Springer, Berlin (1996) Google Scholar
  11. 11.
    Kanerva, P.: Fully distributed representation. In: Proc. 1997 Real World Computing Symposium (RWC’97, Tokyo), Real World Computing Partnership, Tsukuba-City, Japan, pp. 358–365 (1997) Google Scholar
  12. 12.
    Lounesto, P.: Clifford Algebras and Spinors, 2nd edn. Cambridge University Press, Cambridge (2001) zbMATHCrossRefGoogle Scholar
  13. 13.
    Plate, T.: Holographic Reduced Representation: Distributed Representation for Cognitive Structures. CSLI Publications, Stanford (2003) Google Scholar
  14. 14.
    Smolensky, P.: Tensor product variable binding and the representation of symbolic structures in connectionist systems. Artif. Intell. 46, 159–216 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Smolensky, P., Dolan, C.: Tensor product production system: a modular architecture and representation. Connect. Sci. 1(1), 53–68 (1989) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2010

Authors and Affiliations

  1. 1.Faculty of Applied Physics and MathematicsGdańsk University of TechnologyGdańskPoland
  2. 2.Centrum Leo Apostel (CLEA)Vrije Universiteit BrusselBrusselsBelgium

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