Algorithms explained by symmetries

  • Torsten Minkwitz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 900)

Abstract

Many of the linear transforms that are used in digital signalprocessing and other areas have a lot of symmetry properties. This makes the development of fast algorithms for them possible. The usual quadratic cost of multiplying with the matrix is reduced, in some cases to an almost linear complexity. Salient examples are the Fourier transform, the Cosine transform, linear maps with Toepliz matrices or convolutions. The article gives an exact definition of the notion of symmetry that leads to fast algorithms and presents a method to construct those algorithms automatically in the case of an existing symmetry with a soluble group. The results may serve to speed up the multiplication with a transform matrix and also to solve a linear system of equations with symmetry. Even though the construction is done at the level of abstract algebra, the derived algorithms for many linear transforms compare well with the best found in the literature [CoTu65, ElRa82, Ra68, RaYi90]. In most cases, where the new method was applicable, even the manually optimized algorithms [Nu81] were not better, while nothing more than the transform matrix and its symmetry were provided here to optain the results.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ba88]
    BAUM, U.: Schnelle Algorithmen zur Spektraltransformation endlicher Gruppen. Diplomthesis, University of Karlsruhe, August (1988)Google Scholar
  2. [Be84]
    BETH, Th.: Verfahren der schnellen Fourier-Transformation. B. G. Teubner: Stuttgart (1984)Google Scholar
  3. [Be89]
    BETH, TH.: Generating fast Hartley transforms — another application of the algebraic discrete Fourier transform. URSI-ISSSE'89, Erlangen (1989), pp. 688–692Google Scholar
  4. [C188]
    CLAUSEN, M.: Contributions to the design of fast spectral transforms. Habilitation, University of Karlsruhe (1988)Google Scholar
  5. [C189a]
    CLAUSEN, M.: Fast Fourier transforms for metabelian groups. SIAM J. Comput. 18 (1989) No. 3, pp. 584–593Google Scholar
  6. [C189b]
    CLAUSEN, M.: Fast generalized Fourier transforms. Theoret. Comp. Science 67 (1989), pp. 55–63Google Scholar
  7. [ClBa93]
    CLAUSEN, M.-BAUM, U. Fast Fourier Transforms. BI-Wissenschafts-verlag (1993)Google Scholar
  8. [CoTu65]
    COOLEY, J. W.-TUKEY, J. W.: An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19 (1965), pp. 297–301Google Scholar
  9. [CuReSl]
    CURTIS, C. W.-REINER, I.: Methods of Representation theory I. Wiley-Interscience Publ. (1981)Google Scholar
  10. [ElRa82]
    ELLIOT, D. F.-RAO, K. R.: Fast Transforms: Algorithms, Analyses, Applications. Academic Press Inc. (1982)Google Scholar
  11. [StFa79]
    STIEFEL, E.-FÄSSLER, A.: Gruppentheoretische Methoden und ihre Anwendungen. B. G. Teubner: Stuttgart (1979)Google Scholar
  12. [Ja80]
    JACOBSON, N.: Basic Algebra II. W. H. Freeman and Company: New York (1980)Google Scholar
  13. [Ka77]
    KARPOVSKY, M. G.: Fast Fourier transforms on finite non-abelian groups. IEEE Trans. Comput. C-26 (1977), pp. 1028–1030Google Scholar
  14. [KaTr77]
    KARPOVSKY, M. G.-TRACHTENBERG, E. A.: Some optimization problems for convolution systems over finite groups. Information and Control 34 (1977), pp. 227–247Google Scholar
  15. [KaTr79]
    KARPOVSKY, M. G.-TRACHTENBERG, E. A.: Fourier transform over finite groups for error detection and error correction in computation channels. Information and Control 40 (1979), pp. 335–358Google Scholar
  16. [MiCr92]
    MINKWITZ, T.-CREUTZBURG, R.: A New Fast Algebraic Convolution Algorithm. Proceedings of EUSIPCO 6 (1992), pp. 933–936Google Scholar
  17. [Mi93]
    MINKWITZ, T.: Algorithmensynthese für lineare Systeme mit Symmetrie. Doctoral Thesis, University of Karlsruhe (1993)Google Scholar
  18. [Nu81]
    NUSSBAUMER, H. J.: Fast Fourier Transforms and Convolution Algoithms. Springer: Berlin (1981)Google Scholar
  19. [Ra68]
    RADER, C. M.: Discrete Fourier transforms when the number of data samples is prime. Proc. IEEE 56 (1968), pp. 1107–1108Google Scholar
  20. [RaYi90]
    RAO, K. R.-YIP, P.: Discrete Cosine Transform: Algorithms, Advantages, Applications. Academic Press Inc. (1990)Google Scholar
  21. [Se77]
    SERRE, J.-P.: Linear Representations of Finite Groups. Graduate Texts in Mathematics 42, Springer-Verlag (1977)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Torsten Minkwitz
    • 1
  1. 1.Institut für Algorithmen und Kognitive SystemeUniversität KarlsruheDeutschland

Personalised recommendations