Symmetry-based matrix factorization

Abstract We present a method for factoring a given matrix M into a short product of sparse matrices, provided that M has a suitable “symmetry”. This sparse factorization represents a fast algorithm for the matrix–vector multiplication with M. The factorization method consists of two essential steps. First, a combinatorial search is used to compute a suitable symmetry of M in the form of a pair of group representations. Second, the group representations are decomposed stepwise, which yields factorized decomposition matrices and determines a sparse factorization of M. The focus of this article is the first step, finding the symmetries. All algorithms described have been implemented in the library AREP . We present examples for automatically generated sparse factorizations—and hence fast algorithms—for a class of matrices corresponding to digital signal processing transforms including the discrete Fourier, cosine, Hartley, and Haar transforms.

[1]  Markus Püschel,et al.  Automatic generation of fast discrete signal transforms , 2001, IEEE Trans. Signal Process..

[2]  Sebastian Egner,et al.  Zur algorithmischen Zerlegungstheorie linearer Transformationen mit Symmetrie , 1997 .

[3]  J. Wedderburn,et al.  On Hypercomplex Numbers , 1908 .

[4]  D. Rockmore,et al.  Generalized FFT's- A survey of some recent results , 1996, Groups and Computation.

[5]  J. S. Leon,et al.  Permutation Group Algorithms Based on Partitions, I: Theory and Algorithms , 1991, J. Symb. Comput..

[6]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[7]  P. Yip,et al.  Discrete Cosine Transform: Algorithms, Advantages, Applications , 1990 .

[8]  S. Winograd,et al.  Abelian semi-simple algebras and algorithms for the Discrete Fourier Transform , 1984 .

[9]  José M. F. Moura,et al.  Spiral: A Generator for Platform-Adapted Libraries of Signal Processing Alogorithms , 2004, Int. J. High Perform. Comput. Appl..

[10]  Charles M. Rader,et al.  Fast transforms: Algorithms, analyses, applications , 1984 .

[11]  Paul A. Wintz,et al.  Calculation of Fourier transforms on finite Abelian groups (Corresp.) , 1970, IEEE Trans. Inf. Theory.

[12]  Markus Püschel,et al.  Decomposing Monomial Representations of Solvable Groups , 2002, J. Symb. Comput..

[13]  José M. F. Moura,et al.  The Algebraic Approach to the Discrete Cosine and Sine Transforms and Their Fast Algorithms , 2003, SIAM J. Comput..

[14]  Mark G. Karpovsky,et al.  Fast Fourier Transforms on Finite Non-Abelian Groups , 1977, IEEE Transactions on Computers.

[15]  Markus Püschel.  Konstruktive Darstellungstheorie und Algorithmengenerierung , 1998 .

[16]  M. Burrow.  Representation Theory of Finite Groups , 1965 .

[17]  N. Ahmed,et al.  FAST TRANSFORMS, algorithms, analysis, applications , 1983, Proceedings of the IEEE.

[18]  James Stuart Tanton,et al.  Encyclopedia of Mathematics , 2005 .

[19]  David Padua,et al.  Automatic derivation and implementation of signal processing algorithms , 2001, SIGS.

[20]  Dennis M. Healy,et al.  A wreath product group approach to signal and image processing .I. Multiresolution analysis , 2000, IEEE Trans. Signal Process..

[21]  Michael Clausen,et al.  Fast Fourier transforms , 1993 .

[22]  Torsten Minkwitz.  Algorithmensynthese für lineare Systeme mit Symmetrie , 1993 .