Risk hull method for spectral regularization in linear statistical inverse problems

Clément Marteau

ESAIM: Probability and Statistics (2010)

  • Volume: 14, page 409-434
  • ISSN: 1292-8100

Abstract

top
We consider in this paper the statistical linear inverse problem Y = Af + ϵξ where A denotes a compact operator, ϵ a noise level and ξ a stochastic noise. The unknown function f has to be recovered from the indirect measurement Y. We are interested in the following approach: given a family of estimators, we want to select the best possible one. In this context, the unbiased risk estimation (URE) method is rather popular. Nevertheless, it is also very unstable. Recently, Cavalier and Golubev (2006) introduced the risk hull minimization (RHM) method. It significantly improves the performances of the standard URE procedure. However, it only concerns projection rules. Using recent developments on ordered processes, we prove in this paper that it can be extended to a large class of linear estimators.

How to cite

top

Marteau, Clément. "Risk hull method for spectral regularization in linear statistical inverse problems." ESAIM: Probability and Statistics 14 (2010): 409-434. <http://eudml.org/doc/250852>.

@article{Marteau2010,
abstract = { We consider in this paper the statistical linear inverse problem Y = Af + ϵξ where A denotes a compact operator, ϵ a noise level and ξ a stochastic noise. The unknown function f has to be recovered from the indirect measurement Y. We are interested in the following approach: given a family of estimators, we want to select the best possible one. In this context, the unbiased risk estimation (URE) method is rather popular. Nevertheless, it is also very unstable. Recently, Cavalier and Golubev (2006) introduced the risk hull minimization (RHM) method. It significantly improves the performances of the standard URE procedure. However, it only concerns projection rules. Using recent developments on ordered processes, we prove in this paper that it can be extended to a large class of linear estimators. },
author = {Marteau, Clément},
journal = {ESAIM: Probability and Statistics},
keywords = {Inverse problems; oracle inequality; ordered process; risk hull and Tikhonov estimation; inverse problems},
language = {eng},
month = {12},
pages = {409-434},
publisher = {EDP Sciences},
title = {Risk hull method for spectral regularization in linear statistical inverse problems},
url = {http://eudml.org/doc/250852},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Marteau, Clément
TI - Risk hull method for spectral regularization in linear statistical inverse problems
JO - ESAIM: Probability and Statistics
DA - 2010/12//
PB - EDP Sciences
VL - 14
SP - 409
EP - 434
AB - We consider in this paper the statistical linear inverse problem Y = Af + ϵξ where A denotes a compact operator, ϵ a noise level and ξ a stochastic noise. The unknown function f has to be recovered from the indirect measurement Y. We are interested in the following approach: given a family of estimators, we want to select the best possible one. In this context, the unbiased risk estimation (URE) method is rather popular. Nevertheless, it is also very unstable. Recently, Cavalier and Golubev (2006) introduced the risk hull minimization (RHM) method. It significantly improves the performances of the standard URE procedure. However, it only concerns projection rules. Using recent developments on ordered processes, we prove in this paper that it can be extended to a large class of linear estimators.
LA - eng
KW - Inverse problems; oracle inequality; ordered process; risk hull and Tikhonov estimation; inverse problems
UR - http://eudml.org/doc/250852
ER -

References

top
  1. A. Barron, L. Birgé and P. Massart, Risk bounds for model selection via penalization. Probab. Theory Relat. Fields113 (1999) 301–413.  Zbl0946.62036
  2. F. Bauer and T. Hohage, A Lepskij-type stopping rule for regularized Newton methods. Inv. Probab.21 (2005) 1975–1991.  Zbl1091.65052
  3. L. Birgé and P. Massart, Gaussian model selection. J. Eur. Math. Soc.3 (2001) 203–268.  Zbl1037.62001
  4. N. Bissantz, T. Hohage and A. Munk, Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise. Inv. Probab.20 (2004) 1773–1789.  Zbl1077.65060
  5. N. Bissantz, G. Claeskens, H. Holzmann and A. Munk, Testing for lack of fit in inverse regression – with applications to biophotonic imaging. J. R. Stat. Soc. Ser. B71 (2009) 25–48.  Zbl1231.62060
  6. N. Bissantz, T. Hohage, A. Munk and F. Ryumgaart, Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM J. Numer. Anal.45 (2007) 2610–2636.  Zbl1234.62062
  7. Y. Cao and Y. Golubev, On oracle inequalities related to smoothing splines. Math. Meth. Stat.15 (2006) 398–414.  
  8. L. Cavalier and Y. Golubev, Risk hull method and regularization by projections of ill-posed inverse problems. Ann. Statist.34 (2006) 1653–1677.  Zbl1246.62082
  9. L. Cavalier and A.B. Tsybakov, Sharp adaptation for inverse problems with random noise. Probab. Theory Relat. Fields123 (2002) 323–354.  Zbl1039.62031
  10. L. Cavalier, G.K. Golubev, D. Picard and A.B. Tsybakov, Oracle inequalities for inverse problems. Ann. Statist.30 (2002) 843–874.  Zbl1029.62032
  11. D.L. Donoho, Nonlinear solutions of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal.2 (1995) 101–126.  Zbl0826.65117
  12. S. Efromovich, Robust and efficient recovery of a signal passed trough a filter and then contaminated by non-gaussian noise. IEEE Trans. Inf. Theory43 (1997) 1184–1191.  Zbl0881.93081
  13. H.W. Engl, On the choice of the regularization parameter for iterated Tikhonov regularization of ill-posed problems. J. Approx. Theory49 (1987) 55–63.  Zbl0608.65033
  14. H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic Publishers Group, Dordrecht (1996).  Zbl0859.65054
  15. M.S Ermakov, Minimax estimation of the solution of an ill-posed convolution type problem. Probl. Inf. Transm.25 (1989) 191–200.  Zbl0698.62011
  16. Yu. Golubev, The principle of penalized empirical risk in severely ill-posed problems. Theory Probab. Appl.130 (2004) 18–38.  Zbl1064.62011
  17. M. Hanke, Accelerated Lanweber iterations for the solution of ill-posed equations. Numer. Math.60 (1991) 341–373.  Zbl0745.65038
  18. T. Hida, Brownian Motion. Springer-Verlag, New York-Berlin (1980).  Zbl0423.60063
  19. I.M. Johnstone and B.W. Silverman, Speed of estimation in positron emission tomography and related inverse problems. Ann. Statist.18 (1990) 251–280.  Zbl0699.62043
  20. I.M. Johnstone, G. Kerkyacharian, D. Picard and M. Raimondo, Wavelet deconvolution in a periodic setting. J. R. Stat. Soc. B66 (2004) 547–573.  Zbl1046.62039
  21. A. Kneip, Ordered linear smoother. Ann. Statist.22 (1994) 835–866.  Zbl0815.62022
  22. J.M Loubes and C. Ludena, Penalized estimators for non-linear inverse problems. ESAIM: PS14 (2010) 173–191 Zbl1213.62066
  23. C. Marteau, On the stability of the risk hull method for projection estimator. J. Stat. Plan. Inf.139 (2009) 1821–1835.  Zbl1165.62059
  24. P. Mathé, The Lepskij principle revisited. Inv. Probab.22 (2006) L11–L15.  
  25. P. Mathé and S.V. Pereverzev, Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods. SIAM J. Numer. Anal.38 (2001) 1999–2021.  Zbl1049.65046
  26. D.N.G. Roy and L.S. Couchman, Inverse problems and inverse scattering of plane waves. Academic Press, San Diego (2002).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.