Journal of Scientific Computing

, Volume 55, Issue 1, pp 149–172 | Cite as

Fast Approximation of the Discrete Gauss Transform in Higher Dimensions

Article

Abstract

We present a novel approach for the fast approximation of the discrete Gauss transform in higher dimensions. The algorithm is based on the dual-tree technique and introduces a new Taylor series expansion. It compares favorably to existing methods especially when it comes to higher dimensions and a broad range of bandwidths. Numerical results with different datasets in up to 62 dimensions demonstrate its performance.

Keywords

Gauss transform Fast approximation algorithms High-dimensional 

References

  1. 1.
    Ayyagari, V.R., Boughorbel, F., Koschan, A., Abidi, M.A.: A new method for automatic 3D face registration. In: Proc. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recognit., vol. 3, p. 119 (2005) Google Scholar
  2. 2.
    Baxter, B.J.C., Roussos, G.: A new error estimate of the fast Gauss transform. SIAM J. Sci. Comput. 24(1), 257–259 (2002) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Beatson, R., Greengard, L.: A short course on fast multipole methods. In: Wavelets, Multilevel Methods and Elliptic PDEs, pp. 1–37. Oxford University Press, London (1997) Google Scholar
  4. 4.
    Boughorbel, F., Koschan, A., Abidi, M.: A new multi-sensor registration technique for three-dimensional scene modeling with application to unmanned vehicle mobility enhancement. In: Unmanned Ground Veh. Technol. VII (SPIE Conf. Proc.), vol. 5804, pp. 174–181 (2005) Google Scholar
  5. 5.
    Broadie, M., Yamamoto, Y.: Application of the fast Gauss transform to option pricing. Manag. Sci. 49(8), 1071–1088 (2003) MATHCrossRefGoogle Scholar
  6. 6.
    Broadie, M., Yamamoto, Y.: A double-exponential fast Gauss transform algorithm for pricing discrete path-dependent options. Oper. Res. 53(5), 764–779 (2005) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Elgammal, A., Duraiswami, R., Davis, L.S.: Efficient kernel density estimation using the fast Gauss transform with applications to color modeling and tracking. IEEE Trans. Pattern Anal. Mach. Intell. 25(11), 1499–1504 (2003) CrossRefGoogle Scholar
  8. 8.
    François, D., Wertz, V., Verleysen, M.: About the locality of kernels in high-dimensional spaces. In: Proc. Int. Symp. Appl. Stoch. Models Data Anal. (ASMDA), vol. 8, pp. 238–245 (2005) Google Scholar
  9. 9.
    Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Greengard, L., Strain, J.: The fast Gauss transform. SIAM J. Sci. Stat. Comput. 12(1), 79–94 (1991) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Hegland, M., Pestov, V.: Additive models in high dimensions. In: Proc. 12th Comput. Tech. Appl. Conf. (CTAC-2004). ANZIAM J., vol. 46, pp. C1205–C1221 (2005) Google Scholar
  12. 12.
    Keiner, J., Kunis, S., Potts, D.: Fast summation of radial functions on the sphere. Computing 78, 1–15 (2006) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Kunis, S.: Nonequispaced FFT: Generalisation and Inversion. Shaker, Aachen (2006) Google Scholar
  14. 14.
    Lee, D., Gray, A.: Faster Gaussian summation: theory and experiment. In: Proc. 22nd Annu. Conf. Uncertain. Artif. Intell. (UAI-06). AUAI Press, Arlington (2006) Google Scholar
  15. 15.
    Lee, D., Gray, A., Moore, A.: Dual-tree fast Gauss transforms. Adv. Neural Inf. Process. Syst. 18, 747–754 (2006) Google Scholar
  16. 16.
    Matsumoto, M., Nishimura, T.: Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998) MATHCrossRefGoogle Scholar
  17. 17.
    Morariu, V.I., Srinivasan, B.V., Raykar, V.C., Duraiswami, R., Davis, L.S.: Automatic online tuning for fast Gaussian summation. In: Proc. 22nd Annu. Conf. Neural Inf. Process. Syst., pp. 1113–1120. MIT Press, Cambridge (2008) Google Scholar
  18. 18.
    Raykar, V.C., Duraiswami, R.: Very fast optimal bandwidth selection for univariate kernel density estimation. Tech. rep. CS-TR-4774, Dep. Comput. Sci., Univ. Maryland, College Park (2005) Google Scholar
  19. 19.
    Raykar, V.C., Yang, C., Duraiswami, R., Gumerov, N.A.: Fast computation of sums of Gaussians in high dimensions. Tech. rep. CS-TR-4767, Dep. Comput. Sci., Univ. Maryland, College Park (2005) Google Scholar
  20. 20.
    Spivak, M., Veerapaneni, S.K., Greengard, L.: The fast generalized Gauss transform. SIAM J. Sci. Comput. 32(5), 3092–3107 (2010) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Tausch, J., Weckiewicz, A.: Multidimensional fast Gauss transforms by Chebyshev expansions. SIAM J. Sci. Comput. 31, 3547–3565 (2009) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Verleysen, M.: Learning high-dimensional data. In Limit. Future Trends Neural Comput., vol. 186, pp. 141–162 (2003) Google Scholar
  23. 23.
    White, D.A., Jain, R.: Similarity indexing: algorithms and performance. In: Storage and Retr. for Still Image and Video Databases IV, vol. 2670, pp. 62–73 (1996) CrossRefGoogle Scholar
  24. 24.
    Wissel, D.: The discrete Gauss transform—fast approximation algorithms and applications in high dimensions. Diploma thesis. Inst. Numer. Simul., Univ. Bonn (2008) Google Scholar
  25. 25.
    Yang, C., Duraiswami, R., Davis, L.: Efficient kernel machines using the improved fast Gauss transform. In: Adv. Neural Inf. Process. Syst., vol. 17, pp. 1561–1568. MIT Press, Cambridge (2004) Google Scholar
  26. 26.
    Yang, C., Duraiswami, R., Gumerov, N.A., Davis, L.: Improved fast Gauss transform and efficient kernel density estimation. In: Proc. 9th IEEE Int. Conf. Comput. Vis., vol. 1, pp. 664–671 (2003) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institute for Numerical SimulationUniversity of BonnBonnGermany

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