Constrained Optimization Using Geometric Algebra and its Application to Signal Analysis

  • Joan Lasenby
  • Anthony N. Lasenby
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

In this paper we discuss a mathematical system based on the algebras of Grassmann and Clifford [4, 1], called geometric algebra [6]. It is shown how geometric algebra can be used to carry out, in a simple manner, various complex manipulations relevant to matrix-based problems, including that of optimization. In particular we look at how differentiation of certain matrix functions with respect to the matrix, can easily be achieved. The encoding of structure into such problems will be discussed and applied to a multi-source signal separation problem. Other applications are also discussed.

Keywords

Autocorrelation 

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References

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    W.K. Clifford. Applications of Grassmann’s extensive algebra. Am. J. Math. 1: 350–358, 1878.MathSciNetCrossRefMATHGoogle Scholar
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    C. J. L. Doran. Geometric Algebra and its Applications to Mathematical Physics. Ph.D. Thesis, University of Cambridge, 1994.Google Scholar
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    C. J. L. Doran, A. N. Lasenby and S. F. Gull. Geometric Algebra: Applications in Engineering. In W.E. Baylis, editor, Geometric (Clifford) Algebras in Physics. Birkhauser Boston, 65–79, 1996.CrossRefGoogle Scholar
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    H. Grassmann. Der Ort der Hamilton’schen Quaternionen in der Ausdehnungslehre. Math. Ann., 12: 375, 1877.MathSciNetCrossRefMATHGoogle Scholar
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    D. Hestenes. New Foundations for Classical Mechanics. D. Reidel, Dordrecht, 1986.Google Scholar
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    D. Hestenes and G. Sobczyk. Clifford Algebra to Geometric Calculus: A unified language for mathematics and physics. D. Reidel, Dordrecht, 1984.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Joan Lasenby
    • 1
  • Anthony N. Lasenby
    • 2
  1. 1.Department of EngineeringUniversity of CambridgeCambridgeUK
  2. 2.MRAOCavendish LaboratoryCambridgeUK

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