Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-16T11:24:39.688Z Has data issue: false hasContentIssue false
  • English
  • Français

Risk hull method for spectral regularization in linear statistical inverse problems

Published online by Cambridge University Press:  22 December 2010

Clément Marteau
Clément Marteau*
Affiliation:
Institut de Mathématiques, Université de Toulouse, INSA - Département GMM, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France; clement.marteau@insa-toulouse.fr
Get access

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.

Keywords

Inverse problemsoracle inequalityordered processrisk hull and Tikhonov estimation
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 409 - 434
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barron, A., Birgé, L. and Massart, P., Risk bounds for model selection via penalization. Probab. Theory Relat. Fields 113 (1999) 301–413. CrossRef
Bauer, F. and Hohage, T., Lepskij-type, A stopping rule for regularized Newton methods. Inv. Probab. 21 (2005) 1975–1991. CrossRef
Birgé, L. and Massart, P., Gaussian model selection. J. Eur. Math. Soc. 3 (2001) 203–268.
Bissantz, N., Hohage, T. and Munk, A., Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise. Inv. Probab. 20 (2004) 1773–1789. CrossRef
Bissantz, N., Claeskens, G., Holzmann, H. and Munk, A., Testing for lack of fit in inverse regression – with applications to biophotonic imaging. J. R. Stat. Soc. Ser. B 71 (2009) 25–48. CrossRef
Bissantz, N., Hohage, T., Munk, A. and Ryumgaart, F., Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM J. Numer. Anal. 45 (2007) 2610–2636. CrossRef
Cao, Y. and Golubev, Y., On oracle inequalities related to smoothing splines. Math. Meth. Stat. 15 (2006) 398–414.
Cavalier, L. and Golubev, Y., Risk hull method and regularization by projections of ill-posed inverse problems. Ann. Statist. 34 (2006) 1653–1677. CrossRef
Cavalier, L. and Tsybakov, A.B., Sharp adaptation for inverse problems with random noise. Probab. Theory Relat. Fields 123 (2002) 323–354. CrossRef
Cavalier, L., Golubev, G.K., Picard, D. and Tsybakov, A.B., Oracle inequalities for inverse problems. Ann. Statist. 30 (2002) 843–874.
Donoho, D.L., Nonlinear solutions of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 (1995) 101–126. CrossRef
Efromovich, S., Robust and efficient recovery of a signal passed trough a filter and then contaminated by non-gaussian noise. IEEE Trans. Inf. Theory 43 (1997) 1184–1191. CrossRef
Engl, H.W., On the choice of the regularization parameter for iterated Tikhonov regularization of ill-posed problems. J. Approx. Theory 49 (1987) 55–63. CrossRef
H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic Publishers Group, Dordrecht (1996).
Ermakov, M.S, Minimax estimation of the solution of an ill-posed convolution type problem. Probl. Inf. Transm. 25 (1989) 191–200.
Golubev, Yu., The principle of penalized empirical risk in severely ill-posed problems. Theory Probab. Appl. 130 (2004) 18–38.
Hanke, M., Accelerated Lanweber iterations for the solution of ill-posed equations. Numer. Math. 60 (1991) 341–373. CrossRef
T. Hida, Brownian Motion. Springer-Verlag, New York-Berlin (1980).
Johnstone, I.M. and Silverman, B.W., Speed of estimation in positron emission tomography and related inverse problems. Ann. Statist. 18 (1990) 251–280. CrossRef
Johnstone, I.M., Kerkyacharian, G., Picard, D. and Raimondo, M., Wavelet deconvolution in a periodic setting. J. R. Stat. Soc. B 66 (2004) 547–573. CrossRef
Kneip, A., Ordered linear smoother. Ann. Statist. 22 (1994) 835–866. CrossRef
Loubes, J.M and Ludena, C., Penalized estimators for non-linear inverse problems. ESAIM: PS 14 (2010) 173–191 CrossRef
Marteau, C., On the stability of the risk hull method for projection estimator. J. Stat. Plan. Inf. 139 (2009) 1821–1835. CrossRef
Mathé, P., The Lepskij principle revisited. Inv. Probab. 22 (2006) L11–L15. CrossRef
Mathé, P. and Pereverzev, S.V., Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods. SIAM J. Numer. Anal. 38 (2001) 1999–2021. CrossRef
D.N.G. Roy and L.S. Couchman, Inverse problems and inverse scattering of plane waves. Academic Press, San Diego (2002).