Skip to main content
SpringerLink
Log in

Near G-optimal Tchakaloff designs

  • Original paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

We show that the notion of polynomial mesh (norming set), used to provide discretizations of a compact set nearly optimal for certain approximation theoretic purposes, can also be used to obtain finitely supported near G-optimal designs for polynomial regression. We approximate such designs by a standard multiplicative algorithm, followed by measure concentration via Caratheodory-Tchakaloff compression.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Atkinson AK, Donev AN (1992) Optimum experimental designs. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Bloom T, Bos L, Levenberg N, Waldron S (2010) On the convergence of optimal measures. Constr Approx 32:159–179

    Article  MathSciNet  Google Scholar 

  • Bloom T, Bos L, Levenberg N (2011) The asymptotics of optimal designs for polynomial regression, arXiv preprint: arXiv:1112.3735

  • Bos L (1990) Some remarks on the Fejér problem for Lagrange interpolation in several variables. J Approx Theory 60:133–140

    Article  MathSciNet  Google Scholar 

  • Bos L, Calvi JP, Levenberg N, Sommariva A, Vianello M (2011) Geometric weakly admissible meshes, discrete least squares approximations and approximate Fekete points. Math Comput 80:1601–1621

    Article  MathSciNet  Google Scholar 

  • Bos L, Levenberg N, Waldron S (2008) Pseudometrics, distances and multivariate polynomial inequalities. J Approx Theory 153:80–96

    Article  MathSciNet  Google Scholar 

  • Bos L, Vianello M (2012) Low cardinality admissible meshes on quadrangles, triangles and disks. Math Inequal Appl 15:229–235

    MathSciNet  MATH  Google Scholar 

  • Calvi JP, Levenberg N (2008) Uniform approximation by discrete least squares polynomials. J Approx Theory 152:82–100

    Article  MathSciNet  Google Scholar 

  • Caratheodory C (1911) Über den Variabilittsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend Circ Mat Palermo 32:193–217

    Article  Google Scholar 

  • Celant G, Broniatowski M (2017) Interpolation and extrapolation optimal designs 2—finite dimensional general models. Wiley, New York

    Book  Google Scholar 

  • De Castro Y, Gamboa F, Henrion D, Hess R, Lasserre J-B (2019) Approximate optimal designs for multivariate polynomial regressionPalermo. Ann Stat 47:127–155

    Article  Google Scholar 

  • Dette H, Pepelyshev A, Zhigljavsky A (2008) Improving updating rules in multiplicative algorithms for computing D-optimal designs. Comput Stat Data Anal 53:312–320

    Article  MathSciNet  Google Scholar 

  • Dubiner M (1995) The theory of multidimensional polynomial approximation. J Anal Math 67:39–116

    Article  MathSciNet  Google Scholar 

  • Kiefer J, Wolfowitz J (1960) The equivalence of two extremum problems. Can J Math 12:363–366

    Article  MathSciNet  Google Scholar 

  • Kroó A (2011) On optimal polynomial meshes. J Approx Theory 163:1107–1124

    Article  MathSciNet  Google Scholar 

  • Lawson CL, Hanson RJ (1995) Solving least squares problems. Classics in applied mathematics, vol 15. SIAM, Philadelphia

    Book  Google Scholar 

  • Mandal A, Wong WK, Yu Y (2015) Algorithmic searches for optimal designs. In: Dean A, Morris M, Stufken J, Bingham D (eds) Handbook of design and analysis of experiments. Chapman & Hall, New York

    Google Scholar 

  • Piazzon F (2016) Optimal polynomial admissible meshes on some classes of compact subsets of \(\mathbb{R}^d\). J Approx Theory 207:241–264

    Article  MathSciNet  Google Scholar 

  • Piazzon F (2019) Pluripotential numerics. Constr Approx 49:227–263

    Article  MathSciNet  Google Scholar 

  • Piazzon F, Sommariva A, Vianello M (2017a) Caratheodory-Tchakaloff subsampling. Dolomites Res Notes Approx DRNA 10:5–14

    MathSciNet  MATH  Google Scholar 

  • Piazzon F, Sommariva A, Vianello M (2017b) Caratheodory-Tchakaloff least squares. In: Sampling theory and applications. IEEE Xplore Digital Library https://doi.org/10.1109/SAMPTA.2017.8024337

  • Piazzon F, Vianello M (2018) A note on total degree polynomial optimization by Chebyshev grids. Optim Lett 12:63–71

    Article  MathSciNet  Google Scholar 

  • Piazzon F, Vianello M (2019) Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies. Optim Lett 13:1325–1343

    Article  MathSciNet  Google Scholar 

  • Putinar M (1997) A note on Tchakaloff’s theorem. Proc Am Math Soc 125:2409–2414

    Article  MathSciNet  Google Scholar 

  • Sommariva A, Vianello M (2015) Compression of multivariate discrete measures and applications. Numer Funct Anal Optim 36:1198–1223

    Article  MathSciNet  Google Scholar 

  • Sommariva A, Vianello M (2015) Polynomial fitting and interpolation on circular sections. Appl Math Comput 258:410–424

    MathSciNet  MATH  Google Scholar 

  • Sommariva A, Vianello M (2018) Discrete norming inequalities on sections of sphere, ball and torus. J Inequal Spec Funct 9:113–121

    MathSciNet  Google Scholar 

  • Titterington DM (1976) Algorithms for computing D-optimal designs on a finite design space. In: Proceedings of the 1976 conference on information sciences and systems, Baltimora

  • Torsney B, Martin-Martin R (2009) Multiplicative algorithms for computing optimum designs. J Stat Plan Inference 139:3947–3961

    Article  MathSciNet  Google Scholar 

  • Vianello M (2018) Global polynomial optimization by norming sets on sphere and torus. Dolomites Res Notes Approx DRNA 11:10–14

    MathSciNet  Google Scholar 

  • Vianello M (2018) Subperiodic Dubiner distance, norming meshes and trigonometric polynomial optimization. Optim Lett 12:1659–1667

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Verona, Verona, Italy

    Len Bos

  2. Department of Mathematics, University of Padova, Padova, Italy

    Federico Piazzon & Marco Vianello

Authors
  1. Len Bos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Federico Piazzon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marco Vianello
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Vianello.

Additional information

Publisher's Note

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

Work partially supported by the DOR funds and the Project BIRD 181249 of the University of Padova, and by the GNCS-INdAM. This research has been accomplished within the RITA “Research ITalian network on Approximation”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bos, L., Piazzon, F. & Vianello, M. Near G-optimal Tchakaloff designs. Comput Stat 35, 803–819 (2020). https://doi.org/10.1007/s00180-019-00933-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-019-00933-8

Keywords

Search

Navigation