Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
The principle of penalized empirical risk in severely ill-posed problems
Download PDF
Download PDF
  • Published: 05 July 2004

The principle of penalized empirical risk in severely ill-posed problems

  • Yu Golubev1 

Probability Theory and Related Fields volume 130, pages 18–38 (2004)Cite this article

  • 96 Accesses

  • 13 Citations

  • Metrics details

Abstract.

We study a standard method of regularization by projections of the linear inverse problem Y=Ax+ε, where ε is a white Gaussian noise, and A is a known compact operator. It is assumed that the eigenvalues of AA* converge to zero with exponential decay. Such behavior of the spectrum is typical for inverse problems related to elliptic differential operators. As model example we consider recovering of unknown boundary conditions in the Dirichlet problem for the Laplace equation on the unit disk. By using the singular value decomposition of A, we construct a projection estimator of x. The bandwidth of this estimator is chosen by a data-driven procedure based on the principle of minimization of penalized empirical risk. We provide non–asymptotic upper bounds for the mean square risk of this method and we show, in particular, that this approach gives asymptotically minimax estimators in our model example.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Akaike, H.: Information theory and an extension of the maximum likelihood principle. Proc. 2nd Intern. Symp. Inf. Theory, Petrov P.N. and Csaki F. (eds.), Budapest, 1973, pp. 267–281

  2. Barron, A., Birge, L., Massart, P.: Risk bounds for model selection via penalization. Probab. Theory Relat. Fields 113, 301–413 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Belitser, E.N., Levit, Ya, B.: On minimax filtering on ellipsoids. Math. Meth. Statist. 4, 259–273 (1995)

    MathSciNet  MATH  Google Scholar 

  4. Cavalier, L., Golubev, G., Lepski, O., Tsybakov, A: (2003) Block thresholding and sharp adaptive estimation in severely ill-posed inverse problems. Probab. Theory Appl. 3 to appear.

  5. Cavalier, L., Tsybakov, A.: Sharp adaptation for inverse problems with random noise. Probab. Theory Relat. Fields 123, 323–354 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cavalier, L., Golubev, G.K., Picard, D., Tsybakov, A.B.: Oracle inequalities for inverse problems. Ann. Stat. 30 (3), 843–874 (2002)

    MATH  Google Scholar 

  7. Engl, H., Hanke, M., Nuebauer, A.: Regularization of inverse problems. Kluwer academic publishers, 2000

  8. Golubev, G.K., Khasminski, R.Z.: Statistical approach to some inverse boundary problems for partial differential equations. Probl. Inform. Transm. 35 (2), 51–66 (1999)

    MATH  Google Scholar 

  9. Efromovich, S.: Robust and efficient recovery of a signal passed through a filter and then contaminated by non-Gaussian noise. IEEE Trans. Inform. Theory 43, 1184–1191 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ermakov, M.S.: On optimal solutions of the deconvolution problem. Inverse Probl. 6 (5), 863–872 (1990)

    Article  MATH  Google Scholar 

  11. Johnstone, I.M.: Wavelet shrinkage for correlated data and inverse problems: adaptivity results. Statistica Sinica 9, 51–83 (1999)

    MathSciNet  MATH  Google Scholar 

  12. Johnstone, I.M., Silverman, B.W.: Wavelet threshold estimators for data with correlated noise. J. Royal Stat. Soc. Ser. B. 59, 319–351 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kneip, A.: Ordered linear smoothers. Ann. Stat. 22, 835–866 (1994)

    MathSciNet  MATH  Google Scholar 

  14. Landau, H.J., Pollak, H.Q.: Prolate spheroidal wave function, Fourier analysis, and uncertainty – II. Bell System Tech. J. 40, 65–84 (1961)

    MATH  Google Scholar 

  15. Landau, H.J., Pollak, H.Q.: Prolate spheroidal wave function, Fourier analysis, and uncertainty – III. Bell System Tech. J. 41, 1295–1336 (1962)

    MATH  Google Scholar 

  16. Lavrentiev, M.M.: Some improperly posed problems of mathematical physics. Springer Verlag, Berlin Heidelberg New-York, 1967

  17. Mair, B., Ruymgaart, F.H.: Statistical estimation in Hilbert scale. SIAM J. Appl. Math. 56, 1424–1444 (1996)

    MathSciNet  MATH  Google Scholar 

  18. Mallows, C.L.: Some comments on C p . Technometrics 15, 661–675 (1973)

    MATH  Google Scholar 

  19. Pinsker, M.S.: Optimal filtering of square integrable signals in Gaussian white noise. Probl. Inform. Transm. 16, 120–133 (1980)

    MATH  Google Scholar 

  20. Shibata, R.: An optimal selection of regression variables. Boimetrika 68, 45–54 (1981)

    MATH  Google Scholar 

  21. Slepjan, D., Pollak, H.Q.: Prolate spheroidal wave function, Fourier analysis, and uncertainty – I. Bell System Tech. J. 40, 43–64 (1961)

    Google Scholar 

  22. Slepjan, D.: Some asymptotic expansions for prolate spheroidal wave functions. J. Math. Rhys. 44, 99–140 (1965)

    Google Scholar 

  23. Stein, C.M.: Estimation of the mean of a multivariate normal distribution. Ann. Stat. 9, 1135–1151 (1981)

    MathSciNet  MATH  Google Scholar 

  24. Sudakov, V.N., Khalfin, L.A.: Statistical approach to ill-posed problems in mathematical physics. Soviet Mathematics Doklady 157, 1094–1096 (1964)

    MATH  Google Scholar 

  25. Sullivan, F.O’.: A statistical perspective on ill-posed inverse problems. Stat. Sci. 1, 502–527 (1996)

    MATH  Google Scholar 

  26. Tsybakov, A.B.: On the best rate of adaptive estimation in some inverse problems. C.R. Acad. Sci. Paris 330 Série I, 835–840 (2000)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Provence (Aix-Marseille 1), CMI, 39, rue F. Joliot-Curie, 13453, Marseille, Cedex 13, France

    Yu Golubev

Authors
  1. Yu Golubev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu Golubev.

Additional information

Mathematics Subject Classification (2000):Primary 62G05, 62G20; secondary 62C20

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Golubev, Y. The principle of penalized empirical risk in severely ill-posed problems. Probab. Theory Relat. Fields 130, 18–38 (2004). https://doi.org/10.1007/s00440-004-0362-y

Download citation

  • Received: 11 October 2002

  • Revised: 08 March 2004

  • Published: 05 July 2004

  • Issue Date: September 2004

  • DOI: https://doi.org/10.1007/s00440-004-0362-y

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Ill-posed problem
  • Partial differential equation
  • Singular value decomposition
  • Projection estimator
  • Minimax risk
  • Penalization
  • Empirical risk
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature