Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II

Article

Abstract

We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection to the construction of Gel’fand–Tsetlin bases and work in the setting of quivers. We relate this framework to the construction of a configuration space derived from a Bratteli diagram. In this setting the complexity of an algorithm for computing a Fourier transform reduces to the calculation of the dimension of the associated configuration space. Our methods give improved upper bounds for computing the Fourier transform for the general linear groups over finite fields, the classical Weyl groups, and homogeneous spaces of finite groups, while also recovering the best known algorithms for the symmetric group.

Keywords

Generalized Fourier transform Bratteli diagram Path algebra Computational complexity 

Mathematics Subject Classification

43A30 20C15 65T50 

References

  1. 1.
    Auslander, L., Tolimieri, R.: Is computing with the finite Fourier transform pure or applied mathematics? Bull. Am. Math. Soc. (N.S) 1(6), 847–897 (1979)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barros, D., Wilson, S., Kahn, J.: Comparison of orthogonal frequency-division multiplexing and pulse-amplitude modulation in indoor optical wireless links. IEEE Trans. Commun. 60(1), 153–163 (2012)CrossRefGoogle Scholar
  3. 3.
    Bratteli, O.: Inductive limits of finite dimensional \(C^*\)-algebras. Trans. Am. Math. Soc. 171, 195–234 (1972)MathSciNetMATHGoogle Scholar
  4. 4.
    Bürgisser, P., Clausen, M., Shokrollahi, M.: Algebraic Complexity Theory, Volume 315 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1997) (With the collaboration of Thomas Lickteig) Google Scholar
  5. 5.
    Cannon, K., Cariou, R., Chapman, A., Crispin-Ortuzar, M., Fotopoulos, N., Frei, M., Hanna, C., Kara, E., Keppel, D., Liao, L., Privitera, S., Searle, A., Singer, L., Weinstein, A.: Toward early-warning detection of gravitational waves from compact binary coalescence. Astrophys. J. 748(2), 136 (2012)CrossRefGoogle Scholar
  6. 6.
    Clausen, M.: Fast generalized Fourier transforms. Theoret. Comput. Sci. 67(1), 55–63 (1989)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cooley, J., Tukey, J.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Danelakis, A., Mitrouli, M., Triantafyllou, D.: Blind image deconvolution using a banded matrix method. Numer. Algorithms 64(1), 43–72 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Daugherty, Z., Orellana, R.: The quasi-partition algebra. J. Algebra 404, 124–151 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Diaconis, P.: Average running time of the fast Fourier transform. J. Algorithms 1, 187–208 (1980)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Diaconis, P.: Group Representations in Probability and Statistics. In: Lecture Notes—Monograph Series, 11. Institute of Mathematical Statistics, Hayward, CA (1988)Google Scholar
  12. 12.
    Diaconis, P.: A generalization of spectral analysis with application to ranked data. Ann. Stat. 17(3), 949–979 (1989)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Diaconis, P., Rockmore, D.: Efficient computation of the Fourier transform on finite groups. J. Am. Math. Soc. 3(2), 297–332 (1990)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Diaconis, P., Rockmore, D.: Efficient computation of isotypic projections for the symmetric group. In: Groups and computation (New Brunswick, NJ, 1991), Volume 11 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pp. 87–104. Amer. Math. Soc., Providence, RI (1993)Google Scholar
  15. 15.
    Elliott, D., Rao, K.: Fast Transforms: Algorithms, Analyses, Applications. Academic Press Inc. (Harcourt Brace Jovanovich Publishers), New York (1982)MATHGoogle Scholar
  16. 16.
    Elliott, G.: On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. J. Algebra 38(1), 29–44 (1976)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gabriel, P.: Unzerlegbare darstellungen I. Manuscr. Math. 6(1), 71–103 (1972)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gel’fand, I., Cetlin, M.: Finite-dimensional representations of the group of unimodular matrices. Doklady Akad. Nauk SSSR (N.S.) 71, 825–828 (1950)MathSciNetGoogle Scholar
  19. 19.
    Goodman, F., de la Harpe, P., Jones, V.: Coxeter Graphs and Towers of Algebras, Mathematical Sciences Research Institute Publications, vol. 14. Springer, New York (1989)CrossRefGoogle Scholar
  20. 20.
    Grood, C.: The rook partition algebra. J. Comb. Theory Ser. A 113(2), 325–351 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Halverson, T., del Mas, E.: Representations of the Rook–Brauer algebra. Commun. Algebra 42(1), 423–443 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Heideman, M., Johnson, D., Burrus, C.: Gauss and the history of the fast Fourier transform. Arch. Hist. Exact Sci. 34(6), 265–277 (1985)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Humphreys, J.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  24. 24.
    James, G., Kerber, A.: The Representation Theory of the Symmetric Group, Volume 16 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Co., Reading, MA (1981)Google Scholar
  25. 25.
    Johnson, S., Frigo, M.: A modified split-radix FFT with fewer arithmetic operations. IEEE Trans. Signal Process. 55(1), 111–119 (2007)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Leduc, R., Ram, A.: A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: the Brauer, Birman-Wenzl, and type A Iwahori-Hecke algebras. Adv. Math. 125, 1–94 (1997)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Lundy, T., Van Buskirk, J.: A new matrix approach to real FFTs and convolutions of length \(2^k\). Computing 80(1), 23–45 (2007)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Maslen, D.: The efficient computation of Fourier transforms on the symmetric group. Math. Comp. 67(223), 1121–1147 (1998)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Maslen, D., Orrison, M., Rockmore, D.: Computing isotypic projections with the Lanczos iteration. SIAM J. Matrix Anal. Appl. 25(3), 784–803 (2003)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Maslen, D., Rockmore, D.: Adapted diameters and FFTs on groups. In: Proceedings of the 6th ACM-SIAM SODA, pp. 253–262. ACM, New York (1995)Google Scholar
  31. 31.
    Maslen, D., Rockmore, D.: Generalized FFTs—a survey of some recent results. In: Groups and computation, II (New Brunswick, NJ, 1995), Volume 28 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pp. 183–237. Amer. Math. Soc., Providence, RI (1997)Google Scholar
  32. 32.
    Maslen, D., Rockmore, D.: Separation of variables and the computation of Fourier transforms on finite groups. I. J. Am. Math. Soc. 10(1), 169–214 (1997)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Maslen, D., Rockmore, D.: Double coset decompositions and computational harmonic analysis on groups. J. Fourier Anal. Appl. 6(4), 349–388 (2000)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Maslen, D., Rockmore, D.: The Cooley–Tukey FFT and group theory. Not. Am. Math. Soc. 48(10), 1151–1160 (2001)MathSciNetMATHGoogle Scholar
  35. 35.
    Maslen, D., Rockmore, D., Wolff, S.: Separation of variables and the computation of Fourier transforms on semisimple algebras (submitted)Google Scholar
  36. 36.
    Ram, A.: Seminormal representations of Weyl groups and Iwahori–Hecke algebras. Proc. Lond. Math. Soc. (3) 75(1), 99–133 (1997)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Rockmore, D.: Some applications of generalized FFTs. In: Groups and computation, II (New Brunswick, NJ, 1995), Volume 28 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pp. 329–369. Amer. Math. Soc., Providence, RI (1997)Google Scholar
  38. 38.
    Rockmore, D.: The FFT: an algorithm the whole family can use. Comput. Sci. Eng. 2(1), 60–64 (2000)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Rørdam, M., Størmer, E.: Classification of Nuclear \(C^*\)-algebras. In: Entropy in Operator Algebras, Encyclopaedia of Mathematical Sciences, vol. 126. Springer, Berlin (2002)Google Scholar
  40. 40.
    Serre, J.: Linear Representations of Finite Groups. Springer, New York (1977) (Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, vol. 42)Google Scholar
  41. 41.
    Stanley, R.: Differential posets. J. Am. Math. Soc. 1(4), 919–961 (1988)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Stanley, R.: Variations on differential posets. In: Invariant Theory and Tableaux (Minneapolis, MN, 1988), vol. 19 of IMA Vol. Math. Appl., pp. 145–165. Springer, New York (1990)Google Scholar
  43. 43.
    Tolimieri, R., An, M., Lu, C.: Algorithms for Discrete Fourier Transform and Convolution. Signal Processing and Digital Filtering, 2nd edn. Springer, New York (1997)CrossRefMATHGoogle Scholar
  44. 44.
    Van Loan, C.: Computational Frameworks for the Fast Fourier Transform, vol. 10 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1992)Google Scholar
  45. 45.
    Yavne, R.: An economical method for calculating the discrete Fourier transform. Proc. AFIPS Fall Jt. Comput. Conf. 33, 115–125 (1968)Google Scholar
  46. 46.
    Young, A.: On quantitative substitutional analysis. Proc. Lond. Math. Soc. 31(2), 273–288 (1929)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • David Maslen
    • 1
  • Daniel N. Rockmore
    • 2
  • Sarah Wolff
    • 3
  1. 1.HBK Capital ManagementNew YorkUSA
  2. 2.Departments of Mathematics and Computer ScienceDartmouth CollegeHanoverUSA
  3. 3.Department of Mathematics and Computer ScienceDenison UniversityGranvilleUSA

Personalised recommendations