Risk hull method for spectral regularization in linear statistical inverse problems
ESAIM: Probability and Statistics, Tome 14 (2010), pp. 409-434.

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.

DOI : 10.1051/ps/2009011
Classification : 62G05, 62G20
Mots clés : inverse problems, oracle inequality, ordered process, risk hull and Tikhonov estimation
@article{PS_2010__14__409_0,
     author = {Marteau, Cl\'ement},
     title = {Risk hull method for spectral regularization in linear statistical inverse problems},
     journal = {ESAIM: Probability and Statistics},
     pages = {409--434},
     publisher = {EDP-Sciences},
     volume = {14},
     year = {2010},
     doi = {10.1051/ps/2009011},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2009011/}
}
TY  - JOUR
AU  - Marteau, Clément
TI  - Risk hull method for spectral regularization in linear statistical inverse problems
JO  - ESAIM: Probability and Statistics
PY  - 2010
SP  - 409
EP  - 434
VL  - 14
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2009011/
DO  - 10.1051/ps/2009011
LA  - en
ID  - PS_2010__14__409_0
ER  - 
%0 Journal Article
%A Marteau, Clément
%T Risk hull method for spectral regularization in linear statistical inverse problems
%J ESAIM: Probability and Statistics
%D 2010
%P 409-434
%V 14
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2009011/
%R 10.1051/ps/2009011
%G en
%F PS_2010__14__409_0
Marteau, Clément. Risk hull method for spectral regularization in linear statistical inverse problems. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 409-434. doi : 10.1051/ps/2009011. http://archive.numdam.org/articles/10.1051/ps/2009011/

[1] A. Barron, L. Birgé and P. Massart, Risk bounds for model selection via penalization. Probab. Theory Relat. Fields 113 (1999) 301-413. | Zbl

[2] F. Bauer and T. Hohage, A Lepskij-type stopping rule for regularized Newton methods. Inv. Probab. 21 (2005) 1975-1991. | Zbl

[3] L. Birgé and P. Massart, Gaussian model selection. J. Eur. Math. Soc. 3 (2001) 203-268. | Zbl

[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. | Zbl

[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. B 71 (2009) 25-48.

[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.

[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.

[9] L. Cavalier and A.B. Tsybakov, Sharp adaptation for inverse problems with random noise. Probab. Theory Relat. Fields 123 (2002) 323-354. | Zbl

[10] L. Cavalier, G.K. Golubev, D. Picard and A.B. Tsybakov, Oracle inequalities for inverse problems. Ann. Statist. 30 (2002) 843-874. | Zbl

[11] D.L. Donoho, Nonlinear solutions of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 (1995) 101-126. | Zbl

[12] S. Efromovich, 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. | Zbl

[13] H.W. Engl, On the choice of the regularization parameter for iterated Tikhonov regularization of ill-posed problems. J. Approx. Theory 49 (1987) 55-63. | Zbl

[14] H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic Publishers Group, Dordrecht (1996). | Zbl

[15] M.S Ermakov, Minimax estimation of the solution of an ill-posed convolution type problem. Probl. Inf. Transm. 25 (1989) 191-200. | Zbl

[16] Yu. Golubev, The principle of penalized empirical risk in severely ill-posed problems. Theory Probab. Appl. 130 (2004) 18-38. | Zbl

[17] M. Hanke, Accelerated Lanweber iterations for the solution of ill-posed equations. Numer. Math. 60 (1991) 341-373. | Zbl

[18] T. Hida, Brownian Motion. Springer-Verlag, New York-Berlin (1980). | Zbl

[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. | Zbl

[20] I.M. Johnstone, G. Kerkyacharian, D. Picard and M. Raimondo, Wavelet deconvolution in a periodic setting. J. R. Stat. Soc. B 66 (2004) 547-573. | Zbl

[21] A. Kneip, Ordered linear smoother. Ann. Statist. 22 (1994) 835-866. | Zbl

[22] J.M Loubes and C. Ludena, Penalized estimators for non-linear inverse problems. ESAIM: PS 14 (2010) 173-191 | Numdam

[23] C. Marteau, On the stability of the risk hull method for projection estimator. J. Stat. Plan. Inf. 139 (2009) 1821-1835. | Zbl

[24] P. Mathé, The Lepskij principle revisited. Inv. Probab. 22 (2006) L11-L15. | Zbl

[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. | Zbl

[26] D.N.G. Roy and L.S. Couchman, Inverse problems and inverse scattering of plane waves. Academic Press, San Diego (2002).

Cité par Sources :