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From Exponential Analysis to Padé Approximation and Tensor Decomposition, in One and More Dimensions

  • Annie Cuyt
  • Ferre Knaepkens
  • Wen-shin LeeEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11077)

Abstract

Exponential analysis in signal processing is essentially what is known as sparse interpolation in computer algebra. We show how exponential analysis from regularly spaced samples is reformulated as Padé approximation from approximation theory and tensor decomposition from multilinear algebra.

The univariate situation is briefly recalled and discussed in Sect. 1. The new connections from approximation theory and tensor decomposition to the multivariate generalization are the subject of Sect. 2. These connections immediately allow for some generalization of the sampling scheme, not covered by the current multivariate theory.

An interesting computational illustration of the above in blind source separation is presented in Sect. 3.

Keywords

Exponential analysis Parametric method Multivariate Padé approximation Tensor decomposition 

Notes

Acknowledgment

The authors want to thank George Labahn (University of Waterloo, Canada) for making the dataset available to them.

References

  1. 1.
    Allouche, H., Cuyt, A.: Reliable root detection with the qd-algorithm: when Bernoulli, Hadamard and Rutishauser cooperate. Appl. Numer. Math. 60, 1188–1208 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bajzer, Z., Myers, A.C., Sedarous, S.S., Prendergast, F.G.: Padé-Laplace method for analysis of fluorescence intensity decay. Biophys. J. 56(1), 79–93 (1989)CrossRefGoogle Scholar
  3. 3.
    Baker Jr., G., Graves-Morris, P.: Padé Approximants. Encyclopedia of Mathematics and its Applications, vol. 59, 2nd edn. Cambridge University Press, Cambridge (1996)Google Scholar
  4. 4.
    Ben-Or, M., Tiwari, P.: A deterministic algorithm for sparse multivariate polynomial interpolation. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, STOC 1988, pp. 301–309. ACM, New York (1988)Google Scholar
  5. 5.
    Cuyt, A., Lee, W.-s.: Multivariate exponential analysis from the minimal number of samples. Adv. Comput. Math. (2017, to appear)Google Scholar
  6. 6.
    Cuyt, A., Lee, W.-s., Yang, X.: On tensor decomposition, sparse interpolation and Padé approximation. Jaén J. Approx. 8(1), 33–58 (2016)Google Scholar
  7. 7.
    Das, S., Neumaier, A.: Solving overdetermined eigenvalue problems. SIAM J. Sci. Comput. 35(2), A541–A560 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Demeure, C.J.: Fast QR factorization of Vandermonde matrices. Linear Algebra Appl. 122–124, 165–194 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Diederichs, B., Iske, A.: Parameter estimation for bivariate exponential sums. In: IEEE International Conference Sampling Theory and Applications (SampTA 2015), pp. 493–497 (2015)Google Scholar
  10. 10.
    Giesbrecht, M., Labahn, G., Lee, W.-s.: Symbolic-numeric sparse interpolation of multivariate polynomials. In: Proceedings of 2006 International Symposium on Symbolic and Algebraic Computation, ISSAC 2006, pp. 116–123 (2006)Google Scholar
  11. 11.
    Hua, Y., Sarkar, T.K.: Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise. IEEE Trans. Acoust. Speech Sig. Process. 38, 814–824 (1990)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Nyquist, H.: Certain topics in telegraph transmission theory. Trans. Am. Inst. Electr. Eng. 47(2), 617–644 (1928)CrossRefGoogle Scholar
  13. 13.
    Papy, J.M., Lathauwer, L.D., Van Huffel, S.: Exponential data fitting using multilinear algebra: the single-channel and multi-channel case. Numer. Linear Algebra Appl. 12, 809–826 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    de Prony, R.: Essai expérimental et analytique sur les lois de la dilatabilité des fluides élastiques et sur celles de la force expansive de la vapeur de l’eau et de la vapeur de l’alkool, à différentes températures. J. Ec. Poly. 1, 24–76 (1795)Google Scholar
  15. 15.
    Rouquette, S., Najim, M.: Estimation of frequencies and damping factors by two-dimensional ESPRIT type methods. IEEE Trans. Sig. Process. 49(1), 237–245 (2001)CrossRefGoogle Scholar
  16. 16.
    Roy, R., Kailath, T.: ESPRIT-estimation of signal parameters via rotational invariance techniques. IEEE Trans. Acoust., Speech Sig. Process. 37(7), 984–995 (1989)CrossRefGoogle Scholar
  17. 17.
    Shannon, C.E.: Communication in the presence of noise. Proc. IRE 37, 10–21 (1949)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Vervliet, N., Debals, O., Sorber, L., Van Barel, M., De Lathauwer, L.: Tensorlab 3.0, March 2016. https://www.tensorlab.net. Available online

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversiteit Antwerpen (CMI)AntwerpBelgium

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