Skip to main content
SpringerLink
Log in

Perturbations of Christoffel–Darboux Kernels: Detection of Outliers

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

Abstract

Two central objects in constructive approximation, the Christoffel–Darboux kernel and the Christoffel function, encode ample information about the associated moment data and ultimately about the possible generating measures. We develop a multivariate theory of the Christoffel–Darboux kernel in \(\mathbb {C}^d\), with emphasis on the perturbation of Christoffel functions and their level sets with respect to perturbations of small norm or low rank. The statistical notion of leverage score provides a quantitative criterion for the detection of outliers in large data. Using the refined theory of Bergman orthogonal polynomials, we illustrate the main results, including some numerical simulations, in the case of finite atomic perturbations of area measure of a 2D region. Methods of function theory of a complex variable and (pluri) potential theory are widely used in the derivation of our perturbation formulas.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Recall that \(g_\varOmega (z)>0\) for \(z\in \varOmega \), and equal to infinity if \({{\,\mathrm{supp}\,}}(\mu )\) is polar, that is, of logarithmic capacity zero.

  2. This is equivalent to requiring that any subset of multi-indices \(\{ \alpha (0),\alpha (1),\alpha (2),\ldots , \alpha (n) \}\) is downward closed, compare with, e.g., [10].

  3. By definition [20, p. 75], \(\widehat{S}=\{ z\in \mathbb {C}^d: |p(z)|\le \Vert p \Vert _{L^\infty (S)} ~\text{ for } \text{ all } \text{ polynomials }~ p \}\). One easily checks that \(\widehat{S}\) is compact and contains S. Moreover, \(S=\widehat{S}\) for real S by [20, Lemma 5.4.1]. However, unlike the case of one complex variable, \(\widehat{S}\) might be strictly larger than the complement of \(\varOmega \), see, e.g., [1].

  4. The interested reader might want to check that this condition is true for all n if \(\mathcal {N}(\mu )=\mathcal {N}(\nu )\).

References

  1. H. Alexander, J. Wermer, Polynomial Hulls with Convex Fibers, Math. Ann. 271(1985),99-109.

    Article  MathSciNet  Google Scholar 

  2. A. Ambroladze, On Exceptional Sets of Asymptotic Relations for General Orthogonal Polynomials; J. Approx. Theory 82(1995), 257-273.

    Article  MathSciNet  Google Scholar 

  3. T. Bayraktar: Zero distribution of random sparse polynomials, Michigan Math. J. 66(2017), 389-419.

    Article  MathSciNet  Google Scholar 

  4. B. Beckermann, On the numerical condition of polynomial bases: Estimates for the Condition Number of Vandermonde, Krylov and Hankel matrices, Habilitationsschrift, Universität Hannover (1996).

    Google Scholar 

  5. B. Beckermann and N. Stylianopoulos, Bergman orthogonal polynomials and the Grunsky matrix, Constr. Approx. 47(2018) 211-235.

    Article  MathSciNet  Google Scholar 

  6. T. Bloom, N. Levenberg, Weighted pluripotential theory in \(\mathbb{C}^d\), American J. Math.125(2003), 57-103.

    Article  MathSciNet  Google Scholar 

  7. T. Bloom, N. Levenberg, F. Piazzon, F. Wielonsky, Bernstein–Markov: a survey, Dolomite Research Notes on Approximation 8(2015), 75-91.

    MathSciNet  MATH  Google Scholar 

  8. L. Bos, B. Della Vecchia and G. Mastroianni, On the asymptotics of Christoffel functions for centrally symmetric weights functions on the ball in \(\mathbb{R}^n\), Rendiconti del Circolo Matematico di Palermo 52(1998), 277-290.

  9. L. Bos, N. Levenberg, Bernstein–Walsh theory associated to convex bodies and applications to multivariate approximation theory, Computational Methods Function Theory 18(2018), 361-388.

    Article  MathSciNet  Google Scholar 

  10. A. Cohen, G. Migliorati, Optimal weighted least-squares methods. SMAI J. Comput. Math. 3(2017), 181-203.

    Article  MathSciNet  Google Scholar 

  11. A. Delgado, L. Fernandez, T. Perez, M. Pinar, Multivariate Orthogonal Polynomials and Modified Moment Functionals, Sigma 12(2016), 25 pp.

  12. A.M. Delgado, L. Fernández, T.E. Pérez, M.A. Piñar, On the Uvarov modification of two variable orthogonal polynomials on the disk. Complex Anal. Oper. Theory 6 (3)(2012), 665-676.

    Article  MathSciNet  Google Scholar 

  13. C. F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables (second edition), Cambridge Univ. Press, Cambridge, 2014.

    Book  Google Scholar 

  14. G. Freud, Orthogonale Polynome, Birkhäuser, Basel, 1969.

    Book  Google Scholar 

  15. D. Gaier, Lectures on Complex Approximation, Birkhäuser Boston Inc., Boston, MA, 1987.

    Book  Google Scholar 

  16. W. Gautschi, Orthogonal polynomials: computation and approximation, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004.

    Book  Google Scholar 

  17. G.H. Golub, Ch.F. Loan, Matrix computations, 4th edition, Johns Hopkins University Press, Baltimore, 2013.

    MATH  Google Scholar 

  18. B. Gustafsson, M. Putinar, E. Saff, and N. Stylianopoulos, Bergman polynomials on an archipelago: Estimates, zeros and shape reconstruction, Advances in Math. 222 (2009), 1405-1460.

    Article  MathSciNet  Google Scholar 

  19. M. Jarnicki, P. Pflug, First Steps in Several Complex Variables: Reinhardt Domains, Europ. Math. Soc., Zürich, 2008.

    Book  Google Scholar 

  20. M. Klimek, Pluripotential Theory, Clarendon Press, Oxford, 1991.

    MATH  Google Scholar 

  21. A. Kroó and D. S. Lubinsky, Christoffel functions and universality in the bulk for multivariate orthogonal polynomials, Canadian J. Math. 65(2013), 600-620.

    Article  MathSciNet  Google Scholar 

  22. A. Kroó and D. S. Lubinsky, Christoffel functions and universality on the boundary of the ball, Acta Math. Hungarica 140(2013), 117-133.

    Article  MathSciNet  Google Scholar 

  23. J.B. Lasserre, E. Pauwels, Sorting out typicality with the inverse moment matrix SOS polynomial, Proceedings of the 30-th Neural Information Processing Systems Conference, 2016, 15 pp.

  24. J. B. Lasserre, E. Pauwels, The empirical Christoffel function in Statistics and Machine Learning, Advances in Computational Mathematics 45(2019), 1439-1468.

    Article  MathSciNet  Google Scholar 

  25. N. Levenberg, Weighted pluripotential theory results of Berman–Boucksom, arXiv:1010.4035.

  26. D. S. Lubinksy, A new approach to universality limits involving orthogonal polynomials, Ann. Math. 170(2009), 915-939.

    Article  MathSciNet  Google Scholar 

  27. M. Lundin, The extremal psh for the complement of convex, symmetric subsets of \(\mathbb{R}^N\), Michigan Math. J. 32(1985), 197- 201.

  28. V. G. Malyshkin, Multiple-Instance Learning: Radon–Nikodym Approach to Distribution Regression Problem. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.3239043.

  29. V. G. Malyshkin, R. Bakhramov, Mathematical Foundations of Realtime Equity Trading. Liquidity Deficit and Market Dynamics. Automated Trading Machines, SSRN Electronic Journal. https://doi.org/10.2139/ssrn.2679684.

  30. C. Martinez and M.A. Pinar, Orthogonal Polynomials on the Unit Ball and Fourth-Order Partial Differential Equations, SIGMA 12 (2016) 1-11.

    MathSciNet  MATH  Google Scholar 

  31. P. Nevai, Géza Freud, Orthogonal polynomials and Christoffel functions. A case study, J. Approx. Theory, 48 (1986), pp. 3-167.

  32. E. Parzen, On the estimation of a probability density function and mode, Annals of Mathematical Statistics 33(1962), 1065-1076.

    Article  MathSciNet  Google Scholar 

  33. E. Pauwels, F. Bach, J-P. Vert, Relating Leverage Scores and Density using Regularized Christoffel Functions, Advances in Neural Information Processing Systems (NeurIPS) 31, Curran Associates, Inc. (2018), 1670-1679.

    Google Scholar 

  34. W. Plesniak, Siciak’s extremal function in complex and real analysis, Annales Polonici Mathematici 80(2003) 37-46.

    Article  MathSciNet  Google Scholar 

  35. M. Putinar, Spectral analysis of 2D outlier layout, J. Spectral Theory, to appear.

  36. A. Sadullaev, Plurisubharmonic Functions, in: G. M. Khenkin et al. (eds.), Several Complex Variables II, Encyclopaedia of Math. Sci., Springer-Verlag, Berlin Heidelberg 1994.

    Google Scholar 

  37. E. B. Saff, Orthogonal polynomials from a complex perspective, Orthogonal polynomials (Columbus, OH, 1989), Kluwer Acad. Publ., Dordrecht, 1990, 363-393.

  38. E. B. Saff, Remarks on relative asymptotics for general orthogonal polynomials, Recent trends in orthogonal polynomials and approximation theory, Contemp. Math., vol. 507, Amer. Math. Soc., Providence, RI, 2010, pp. 233-239.

  39. E.B. Saff, V. Totik, Logarithmic Potentials with External Fields, Grundlehren der mathematischen Wissenschaften, vol. 316, Springer, Heidelberg (1997).

    Book  Google Scholar 

  40. B. Simon, Orthogonal Polynomials on the Unit Circle, Part 2: Spectral theory, Amer. Math. Soc., Providence, RI, 2009.

    Google Scholar 

  41. B. Simon, The Christoffel-Darboux kernel, in vol. “Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A volume in Honor of Vladimir G. Mazya’s 70th Birthday”, (D. Mitrea, M. Mitrea, eds.), Proc. Symp. Pure Math. Amer. Math. Soc., Providence, R. I., 2008, pp. 314–355.

  42. B. Simon, Weak convergence of CD kernels and applications, Duke Math. J. 146(2009), 305-330.

    Article  MathSciNet  Google Scholar 

  43. H. Stahl, V. Totik, General Orthogonal Polynomials, Cambridge Univ. Press, Cambridge, 1992.

    Book  Google Scholar 

  44. P. K. Suetin, Polynomials orthogonal over a region and Bieberbach polynomials, Proceedings of the Steklov Institute of Mathematics, 100 (1971), Amer. Math. Soc., Providence, R.I., 1974.

    Google Scholar 

  45. K. C. Toh, L. N. Trefethen, The Kreiss matrix theorem on a general complex domain, SIAM J. Matrix Anal. Appl. 21 (1999), 145–165.

    Article  MathSciNet  Google Scholar 

  46. V. Totik, Asymptotics for Christoffel functions for general measures on the real line, J. Analyse Math. 81 (2000), 283-303.

    Article  MathSciNet  Google Scholar 

  47. V. Totik, Christoffel functions on curves and domains, Trans. Amer. Math. Soc. 362 (2010), no. 4, 2053–2087.

    Article  MathSciNet  Google Scholar 

  48. V. N. Vapnik, Statistical Learning Theory, Wiley Interscience, New York, 1998.

    MATH  Google Scholar 

  49. Y. Xu, Orthogonal polynomials of several variables, Ch. 2 of Multivariable Special Functions, Cambridge Univ. Press, to appear; arxiv:1701.02709.

  50. Y. Xu, Asymptotics for orthogonal polynomials and Christoffel functions on a ball, Methods Appl. Analysis 3(1996), 257-272.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the Mathematical Research Institute at Oberwolfach, Germany, which provided exceptional working conditions for a Research in Pairs collaboration in 2016 during which time the main ideas of this manuscript were developed. Special thanks are due to the referees for a careful examination of the manuscript and authoritative comments which led to a clarification of Lemma 2.8.

Author information

Authors and Affiliations

  1. Department of Mathematics, Université de Lille, 59655, Villeneuve d’Ascq, France

    Bernhard Beckermann

  2. Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA, 93106-3080, USA

    Mihai Putinar

  3. School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK

    Mihai Putinar

  4. Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA

    Edward B. Saff

  5. Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678, Nicosia, Cyprus

    Nikos Stylianopoulos

Authors
  1. Bernhard Beckermann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mihai Putinar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edward B. Saff
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nikos Stylianopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mihai Putinar.

Additional information

Communicated by PETRUSHEV Pencho.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author was supported in part by the Labex CEMPI (ANR- 11-LABX-0007-01). The third author was partially supported by the US National Science Foundation grant DMS-1516400. The forth author was supported by the University of Cyprus Grant 3/311-21027.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beckermann, B., Putinar, M., Saff, E.B. et al. Perturbations of Christoffel–Darboux Kernels: Detection of Outliers. Found Comput Math 21, 71–124 (2021). https://doi.org/10.1007/s10208-020-09458-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-020-09458-9

Keywords

Mathematics Subject Classification

Search

Navigation