Advertisement

Sparse polynomial interpolation: sparse recovery, super-resolution, or Prony?

  • Cédric Josz
  • Jean Bernard LasserreEmail author
  • Bernard Mourrain
Article
  • 8 Downloads

Abstract

We show that the sparse polynomial interpolation problem reduces to a discrete super-resolution problem on the n-dimensional torus. Therefore, the semidefinite programming approach initiated by Candès and Fernandez-Granda (Commun. Pure Appl. Math. 67(6) 906–956, 2014) in the univariate case can be applied. We extend their result to the multivariate case, i.e., we show that exact recovery is guaranteed provided that a geometric spacing condition on the supports holds and evaluations are sufficiently many (but not many). It also turns out that the sparse recovery LP-formulation of 1-norm minimization is also guaranteed to provide exact recovery provided that the evaluations are made in a certain manner and even though the restricted isometry property for exact recovery is not satisfied. (A naive sparse recovery LP approach does not offer such a guarantee.) Finally, we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition. We provide two sets of numerical experiments, one in which the super-resolution technique and Prony’s method seem to cope equally well with noise, and another in which the super-resolution technique seems to cope with noise better than Prony’s method, at the cost of an extra computational burden (i.e., a semidefinite optimization).

Keywords

Linear programming Prony’s method Semidefinite programming super-resolution 

Mathematics Subject Classification (2010)

90-08 90C22 90C25 65K05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The research of the second author was funded by the European Research Council (ERC) under the European’s Union Horizon 2020 research and innovation program (grant agreement 666981 TAMING).

Funding information

The work of the first two authors was funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement 666981 TAMING).

References

  1. 1.
    Azaïs, J.-M., de Castro, Y., Gamboa, F.: Spike detection from inaccurate samplings. Appl. Comput. Harmon. Anal. 38(2), 177–195 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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, pp. 301–309. ACM (1988)Google Scholar
  3. 3.
    Berlekamp, E.R.: Nonbinary BCH decoding. IEEE Trans. Inf. Theory 14(2), 242–242 (1968)CrossRefGoogle Scholar
  4. 4.
    Beylkin, G., Monzón, L.: On approximation of functions by exponential sums. Appl. Comput. Harmon. Anal. 19(1), 17–48 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Candès, E.J.: The restricted isometry property and its implications for compressed sensing. C.R. Acad. Sci. Paris Ser. I 346, 589–592 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Candès, E.J., Carlos, F.-G.: Super-resolution from noisy data. J. Fourier Anal. Appl. 19(6), 1229–1254 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Candès, E.J., Fernandez-Granda, C.: Towards a mathematical theory of super-resolution. Commun. Pure Appl. Math. 67(6), 906–956 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Candes, E.J., Plan, Y.: A probabilistic and RIPless theory of compressed sensing. IEEE Trans. Inf. Theory 57(11), 7235–7254 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Candes, E.J., Tao, T.: Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Candès, E.J., Tao, T.: Decoding by linear programming. IEEE Inform. Theory 51(12), 4203–4215 (2014). 2005MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Candès, E.J., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Curto, R.E., Fialkow, L.A.: Truncated K-moment problems in several variables. J. Operator Theory 54, 189–226 (2005)MathSciNetzbMATHGoogle Scholar
  13. 13.
    de Baron de Prony, G.R.: Essai expérimental et analytique: Sur les lois de la dilatabilité de fluides élastique et sur celles de la force expansive de la vapeur de l’alcool, à différentes températures. J. Ecole Polyt. 1, 24–76 (1795)Google Scholar
  14. 14.
    Cuyt, A., Lee, W.-S.: Sparse interpolation and rational approximation. In: Hardin, D., Lubinsky, D., Simanek, B., et al. (eds.) Contemporary Mathematics, vol. 661, pp 229–242. American Mathematical Society, Providence (2016)Google Scholar
  15. 15.
    De Castro, Y., Gamboa, F., Henrion, D., Lasserre, J.-B.: Exact solutions to super-resolution on semi-algebraic domains in higher dimensions. IEEE Trans. Inform. Theory 63, 621–630 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dantzig, G.B., Thapa, M.N.: Linear programming 1: Introduction. Springer-verlag New York Inc., New York (1997)zbMATHGoogle Scholar
  17. 17.
    Duval, V., Peyré, G.: Exact support recovery for sparse spikes deconvolution. Found. Comput. Math. 15(5), 1315–1355 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fan, Y.Y., Kamath, C.: A comparison of compressed sensing and sparse recovery algorithms applied to simulation data. Stat. Optim. Inform. Computing 4, 194–213 (2016)MathSciNetGoogle Scholar
  19. 19.
    Filbir, F., Schröder, K.: Exact recovery of discrete measures from Wigner D-moments. arXiv:1606.05306 (2016)
  20. 20.
    Giesbrecht, M., Labahn, G., Lee, W.-S.: Symbolic–numeric sparse interpolation of multivariate polynomials. J. Symb. Comput. 44(8), 943–959 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Golub, G., Pereyra, V.: Separable nonlinear least squares: The variable projection method and its applications. Inverse Prob. 19(2), R1–R26 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grigoriev, D.Y., Karpinski, M., Singer, M.F.: Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM J. Comput. 19 (6), 1059–1063 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Harmouch, J., Khalil, H., Mourrain, B.: Structured Low Rank Decomposition of Multivariate Hankel Matrices. Linear Algebra and its Applications (2017)Google Scholar
  24. 24.
    Hassanieh, Hitham, Indyk, Piotr, Katabi, Dina, Price, Eric: Nearly optimal sparse Fourier transform. In: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, STOC ’12, pp. 563–578. ACM Press (2012)Google Scholar
  25. 25.
    Josz, C., Molzahn, D.K.: Large Scale Complex Polynomial Optimization. arXiv:1508.02068
  26. 26.
    Kaltofen, E., Lakshman, Y.N.: Sparse multivariate polynomial interpolation algorithms. In: Proceedings of the International Symposium ISSAC’88 on Symbolic and Algebraic Computation, ISSAC ’88, pp 467–474. Springer, London (1989)Google Scholar
  27. 27.
    Kaltofen, E.L., Lee, W.-S., Yang, Z.: Fast Estimates of Hankel Matrix Condition Numbers and Numeric Sparse Interpolation, pp 130–136. ACM Press, New York (2011)zbMATHGoogle Scholar
  28. 28.
    Krasovskii, N.N.: Theory of Motion Control. Moscow, Nauka (1968). (in Russian)Google Scholar
  29. 29.
    Kunis, S., Peter, T., Römer, T., von der Ohe, U.: A multivariate generalization of Prony’s method. Linear Algebra Appl. 490, 31–47 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Laurent, M., Mourrain, B.: A generalized flat extension theorem for moment matrices. Arch. Math. 93(1), 87–98 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Massey, J.: Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory 15(1), 122–127 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mourrain, B.: Polynomial-exponential decomposition from moments. Found. Comput. Math. 18(6), 1435–1492 (2018).  https://doi.org/10.1007/s10208-017-9372-x MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Neustadt, L.W.: Optimization, a moment problem, and nonlinear programming. Journal of the Society for Industrial and Applied Mathematics Series A Control 2(1), 33–53 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Nie, J.: Optimality conditions and finite convergence of Lasserre?s hierarchy. Math. Program. Ser Optimality A 146(1-2), 97–121 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Pereyra, V., Scherer, G., et al.: Exponential Data Fitting and Its Applications. Bentham Science Publishers, Sharjah (2012)CrossRefGoogle Scholar
  36. 36.
    Poon, C., Peyré, G.: Multi-dimensional sparse super-resolution. SIAM J. Math. Anal. 51(1), 1–44 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Potts, D., Tasche, M.: Nonlinear approximation by sums of nonincreasing exponentials. Appl. Anal. 90(3-4), 609–626 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Richard, R., Kailath, T.: ESPRIT-Estimation of signal parameters via rotational invariance techniques. IEEE Trans. Acoust. Speech Signal Process. 37(7), 984–995 (1989)CrossRefGoogle Scholar
  39. 39.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill Education, New York (1986)zbMATHGoogle Scholar
  40. 40.
    Sauer, T.: Prony’s method in several variables. Numer. Math. 136(2), 411–438 (2017).  https://doi.org/10.1007/s00211-016-0844-8 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Stoica, P., Moses, R.L.: Spectral Analysis of Signals. Pearson/prentice Hall, Upper Saddle River (2005)Google Scholar
  42. 42.
    Lee Swindlehurst, A., Kailath, T.: A performance analysis of subspace-based methods in the presence of model errors. I. The MUSIC algorithm. IEEE Trans. Signal Process. 40(7), 1758–1774 (1992)CrossRefzbMATHGoogle Scholar
  43. 43.
    Zippel, R.: Probabilistic algorithms for sparse polynomials. In: Proceedings of the International Symposiumon on Symbolic and Algebraic Computation, EUROSAM ’79, pp 216–226. Springer, London (1979)Google Scholar
  44. 44.
    Zippel, R.: Interpolating polynomials from their values. J. Symb. Comput. 9 (3), 375–403 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Iohvidov, I.S.: Hankel and Toeplitz Matrices and Forms: Algebraic Theory. Birkhäuser Verlag, Boston (1982)zbMATHGoogle Scholar
  46. 46.
    Comer, M.T., Kaltofen, E.L., Pernet, C.: Sparse polynomial interpolation and Berlekamp/Massey algorithms that correct outlier errors in input values. In: Proceedings of the Sparse Polynomial International Symposium on Symbolic and Algebraic Computation, ISSAC ’12, pp. 138–145. Grenoble, France. 2012 (2012)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.LAAS-CNRSToulouse Cédex 4France
  2. 2.LAAS-CNRS and Institute of MathematicsToulouse Cédex 4France
  3. 3.Université Côte d’Azur, InriaSophia AntipolisFrance

Personalised recommendations