ABSTRACT
This paper introduces new techniques for the efficient computation of discrete Fourier transforms (DFTs) of Sn-k-invariant functions on the symmetric group Sn. We uncover diamond- and leaf-rake-like structures in Young's seminormal and orthogonal representations. Combining this with both a multiresolution scheme and an anticipation technique for saving scalar multiplications leads to linear time partial FFTs. Following the inductive version of Young's branching rule we obtain a global FFT that computes a DFT of Sn-k-invariant functions on Sn in at most ck...[Sn : Sn-k] scalar multiplications and additions, where ck denotes a positive constant depending only on k. This run-time, which is linear in [Sn : Sn-k], is order optimal and improves Maslen's algorithm. For example, it takes less than one second on a standard notebook to run our FFT algorithm for an Sn-2-invariant real-valued function on Sn, n=5000.
- U. Baum and M. Clausen. Some lower and upper complexity bounds for generalized Fourier transforms and their inverses. SIAM J. Comput., 20(3):451--459, 1991. Google ScholarDigital Library
- P. Bürgisser, M. Clausen, and M. A. Shokrollahi. Algebraic Complexity Theory, volume 315 of Grundlehren der mathematischen Wissenschaften. Springer, 1997.Google Scholar
- M. Clausen. Fast generalized Fourier transforms. Theor. Comput. Sci., 67(1):55--63, 1989. Google ScholarDigital Library
- M. Clausen and U. Baum. Fast Fourier transforms for symmetric groups: theory and implementations. Math. Comp., 61(204):833--847, 1993.Google ScholarCross Ref
- M. Clausen and R. Kakarala. Computing Fourier transforms and convolutions of $S_n-1$-invariant signals on $S_n$ in time linear in n. Appl. Math. Lett., 23(2):183--187, 2010.Google ScholarCross Ref
- R. L. Graham, D. E. Knuth, and O. Patashnik. Concrete Mathematics: A Foundation for Computer Science, second edition. Addison-Wesley, 1994. Google ScholarDigital Library
- P. C. Hühne. Beiträge zum Entwurf größenoptimaler schneller Fouriertransformationen auf gewissen homogenen R\"aumen symmetrischer Gruppen. PhD thesis, Universit\"at Bonn, 2016.Google Scholar
- G. James and A. Kerber. The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and its Applications, 16. Addison-Wesley, 1981.Google Scholar
- R. Kondor. A Fourier space algorithm for solving quadratic assignment problems. In SODA 2010, pages 1017--1028, 2010. Google ScholarDigital Library
- R. Kondor and M. S. Barbosa. Ranking with kernels in Fourier space. In COLT 2010, pages 451--463, 2010.Google Scholar
- R. Kondor and W. Dempsey. Multiresolution analysis on the symmetric group. In NIPS 2012, pages 1646--1654, 2012. Google ScholarDigital Library
- R. Kondor, A. Howard, and T. Jebara. Multi-object tracking with representations of the symmetric group. In AISTATS 2007, pages 211--218, 2007.Google Scholar
- R. Kondor, N. Shervashidze, and K. M. Borgwardt. The graphlet spectrum. In ICML 2009, pages 529--536, 2009. Google ScholarDigital Library
- D. K. Maslen. The efficient computation of Fourier transforms on the symmetric group. Math. Comput., 67(223):1121--1147, 1998. Google ScholarDigital Library
- D. K. Maslen and D. N. Rockmore. Generalized FFT's -- A survey of some recent results. In Groups and Computation, Proceedings of a DIMACS Workshop, New Brunswick, New Jersey, USA, June 7--10, 1995, pages 183--238, 1995.Google Scholar
- D. Rockmore, P. Kostelec, W. Hordijk, and P. F. Stadler. Fast Fourier transforms for fitness landscapes. Applied and Computational Harmonic Analysis, 11(1):57--76, 2002.Google ScholarCross Ref
- J.-P. Serre. Linear Representations of Finite Groups, volume 42 of Graduate Texts in Mathematics. Springer, 1977.Google Scholar
- N. Sloane. The on-line encyclopedia of integer sequences. https://oeis.org.Google Scholar
Index Terms
- Linear Time Fourier Transforms of Sn-k-invariant Functions on the Symmetric Group Sn
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