Skip to main content

Fast Interpolation and Fourier Transform in High-Dimensional Spaces

Part of the Advances in Intelligent Systems and Computing book series (AISC,volume 857)

Abstract

In scientific computing, the problem of finding an analytical representation of a given function \(f: \Omega \subseteq \mathbb {R}^m \longrightarrow \mathbb {R},\mathbb {C}\) is ubiquitous. The most practically relevant representations are polynomial interpolation and Fourier series. In this article, we address both problems in high-dimensional spaces. First, we propose a quadratic-time solution of the Multivariate Polynomial Interpolation Problem (PIP), i.e., the N(mn) coefficients of a polynomial Q, with \(\deg (Q)\le n\), uniquely fitting f on a determined set of generic nodes \(P\subseteq \mathbb {R}^m\) are computed in \(\mathcal {O}(N(m,n)^2)\) time requiring storage in \(\mathcal {O}(mN(m,n))\). Then, we formulate an algorithm for determining the N(mn) Fourier coefficients with positive frequency of the Fourier series of f up to order n in the same amount of computational time and storage. Especially in high dimensions, this provides a fast Fourier interpolation, outperforming modern Fast Fourier Transform methods. We expect that these fast and scalable solutions of the polynomial and Fourier interpolation problems in high-dimensional spaces are going to influence modern computing techniques occurring in Big Data and Data Mining, Deep Learning, Image and Signal Analysis, Cryptography, and Non-linear Optimization.

Keywords

  • Fast Fourier transform
  • (Multivariate) Polynomial interpolation
  • Signal analysis
  • Gradient descent
  • Optimization
  • Newton-Raphson iteration
  • Integration of multivariate functions
  • Big data
  • Machine & deep learning
  • Data mining

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-01177-2_5
  • Chapter length: 23 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   219.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-01177-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   279.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.

References

  1. Bishop, C.M.: Pattern Recognition and Machine Learning. Springer, New York (2006)

    MATH  Google Scholar 

  2. Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press (2016)

    Google Scholar 

  3. Pal, A., Pal, S.K.: Pattern Recognition and Big Data. World Scientific (2016)

    Google Scholar 

  4. Chu, E., George, A.: Inside the FFT black box: serial and parallel fast Fourier transform algorithms. CRC Press (1999)

    Google Scholar 

  5. Rockmore, D.N.: The FFT: an algorithm the whole family can use. Comput. Sci. Eng. 2(1), 60–64 (2000)

    CrossRef  Google Scholar 

  6. Edwards, K.J., Gaber, M.M.: Astronomy and Big Data: A Data Clustering Approach to Identifying Uncertain Galaxy Morphology. Springer, Heidelberg (2014)

    CrossRef  Google Scholar 

  7. Kremer, J., Stensbo-Smidt, K., Gieseke, F., Pedersen, K.S., Igel, C.: Big universe, big data: machine learning and image analysis for astronomy. IEEE Intell. Syst. 32(2), 16–22 (2017)

    CrossRef  Google Scholar 

  8. Altaf-Ul-Amin, M., Afendi, F.M., Kiboi, S.K., Kanaya, S.: Systems biology in the context of big data and networks. In: BioMed Research International, vol. 2014 (2014)

    Google Scholar 

  9. Howe, D., Costanzo, M., Fey, P., Gojobori, T., Hannick, L., Hide, W., Hill, D.P., Kania, R., Schaeffer, M., St. Pierre, S. et al.: Big data: the future of biocuration. Nature 455(7209), 47–50 (2008)

    Google Scholar 

  10. Telenti, A., Pierce, L.C., Biggs, W.H., di Iulio, J., Wong, E.H., Fabani, M.M., Kirkness, E.F., Moustafa, A., Shah, N., Xie, C., et al.: Deep sequencing of 10,000 human genomes. In: Proceedigns of National Academy of Science, USA (2016). 201613365

    Google Scholar 

  11. Walker, S.J.: Big data: a revolution that will transform how we live, work, and think (2014)

    Google Scholar 

  12. Nunan, D., Di Domenico, M.: Market research & the ethics of big data. Intl. J. Market Res. 55(4), 505–520 (2013)

    CrossRef  Google Scholar 

  13. Hecht, M., Cheeseman, B.L., Hoffmann, K.B., Sbalzarini, I.F.: A quadratic-time algorithm for general multivariate polynomial interpolation. arXiv Preprint DOI arXiv:1710.10846. math.NA

  14. Strassen, V.: Gaussian elimination is not optimal. Numerische mathematik 13(4), 354–356 (1969)

    MathSciNet  CrossRef  Google Scholar 

  15. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)

    MathSciNet  CrossRef  Google Scholar 

  16. Beard, J.: The FFT in the 21st century: Eigenspace processing. Springer, Heidelberg (2013)

    Google Scholar 

  17. Olver, P.J.: On multivariate interpolation. School of Mathematics, University of Minnesota, Minnesota (2009)

    Google Scholar 

  18. Yao, T.H., Chung, K.C.: On lattices admitting unique Lagrange interpolations. SIAM J. Numer. Anal. 14(4), 735–743 (1977). https://doi.org/10.14658/pupj-drna-2015-Special_Issue-4

    MathSciNet  CrossRef  MATH  Google Scholar 

  19. Meijering, E.: A chronology of interpolation: from ancient astronomy to modern signal and image processing. Proc. IEEE 90(3), 319–342 (2002)

    CrossRef  Google Scholar 

  20. Stoer, J., Bauer, F.L., Bulirsch, R.: Numerische Mathematik, vol. 4. Springer, Heidelberg (1989)

    Google Scholar 

  21. Schrader, B., Reboux, S., Sbalzarini, I.F.: Discretization correction of general integral PSE operators for particle methods. J. Comput. Phys. 229(11), 4159–4182 (2010). https://doi.org/10.1016/j.jcp.2010.02.004

    MathSciNet  CrossRef  MATH  Google Scholar 

  22. Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 1979. [Online]. Available: http://doi.acm.org/10.1145/359168.359176

  23. Hanzon, B., Jibetean, D.: Global minimization of a multivariate polynomial using matrix methods. J. Global Optim. 27(1), pp. 1–23 (2003). http://dx.doi.org/10.1023/A:1024664432540

  24. Parrilo, P.A., Sturmfels, B.: Minimizing polynomial functions. Algorithmic and quantitative real algebraic geometry, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 60, 83–99 (2003)

    Google Scholar 

  25. Milnor, J.: Morse theory. In: Annals of Mathematics Studies, vol. 51. Princeton University Press (1963)

    Google Scholar 

  26. Asmus, J., Müller, C.L., Sbalzarini, I.F.: Lp-adaptation: simultaneous design centering and robustness estimation of electronic and biological systems. Sci. Rep. 7 (2017)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michael Hecht .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Hecht, M., Sbalzarini, I.F. (2019). Fast Interpolation and Fourier Transform in High-Dimensional Spaces. In: Arai, K., Kapoor, S., Bhatia, R. (eds) Intelligent Computing. SAI 2018. Advances in Intelligent Systems and Computing, vol 857. Springer, Cham. https://doi.org/10.1007/978-3-030-01177-2_5

Download citation