Adaptive hard-thresholding for linear inverse problems
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 485-499.

A number of regularization methods for discrete inverse problems consist in considering weighted versions of the usual least square solution. These filter methods are generally restricted to monotonic transformations, e.g. the Tikhonov regularization or the spectral cut-off. However, in several cases, non-monotonic sequences of filters may appear more appropriate. In this paper, we study a hard-thresholding regularization method that extends the spectral cut-off procedure to non-monotonic sequences. We provide several oracle inequalities, showing the method to be nearly optimal under mild assumptions. Contrary to similar methods discussed in the literature, we use here a non-linear threshold that appears to be adaptive to all degrees of irregularity, whether the problem is mildly or severely ill-posed. Finally, we extend the method to inverse problems with noisy operator and provide efficiency results in a conditional framework.

DOI : 10.1051/ps/2012003
Classification : 62G08, 62J02
Mots-clés : inverse problems, singular value decomposition, hard-thresholding 
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Rochet, Paul. Adaptive hard-thresholding for linear inverse problems. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 485-499. doi : 10.1051/ps/2012003. http://archive.numdam.org/articles/10.1051/ps/2012003/

[1] F. Abramovich and B.W. Silverman, Wavelet decomposition approaches to statistical inverse problems. Biometrika 85 (1998) 115-129. | MR | Zbl

[2] N. Bissantz, T. Hohage, A. Munk and F. Ruymgaart, Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM J. Numer. Anal. 45 (2007) 2610-2636 (electronic). | MR | Zbl

[3] L. Cavalier, Nonparametric statistical inverse problems. Inverse Problems 24 (2008) 034004. | MR | Zbl

[4] L. Cavalier and G.K. Golubev, Risk hull method and regularization by projections of ill-posed inverse problems. Ann. Statist. 34 (2006) 1653-1677. | MR | Zbl

[5] L. Cavalier and N.W. Hengartner, Adaptive estimation for inverse problems with noisy operators. Inverse Problems 21 (2005) 1345-1361. | MR | Zbl

[6] L. Cavalier, G.K. Golubev, D. Picard and A.B. Tsybakov, Oracle inequalities for inverse problems. Ann. Statist. 30 (2000) 843-874. | MR | Zbl

[7] S. Efromovich and V. Koltchinskii, On inverse problems with unknown operators. IEEE Trans. Inform. Theory 47 (2001) 2876-2894. | MR | Zbl

[8] H.W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems, Math. Appl., vol. 375. Kluwer Academic Publishers Group, Dordrecht (1996). | MR | Zbl

[9] P.C. Hansen, The truncated SVD as a method for regularization. BIT 27 (1987) 534-553. | MR | Zbl

[10] P.C. Hansen and D.P. O'Leary. The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14 (1993) 1487-1503. | MR | Zbl

[11] M. Hoffmann and M. Reiss, Nonlinear estimation for linear inverse problems with error in the operator. Ann. Statist. 36 (2008) 310-336. | MR | Zbl

[12] J.M. Loubes, l1 penalty for ill-posed inverse problems. Comm. Statist. Theory Methods 37 (2008) 1399-1411. | MR | Zbl

[13] J.M. Loubes and C. Ludeña, Adaptive complexity regularization for linear inverse problems. Electron. J. Stat. 2 (2008) 661-677. | MR

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

[15] F. Natterer, The mathematics of computerized tomography, Class. Appl. Math., vol. 32. SIAM, Philadelphia, PA (2001). Reprint of the 1986 original. | MR | Zbl

[16] J.A. Scales and A. Gersztenkorn, Robust methods in inverse theory. Inverse Problems 4 (1988) 1071-1091. | MR | Zbl

[17] C.M. Stein, Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 (1981) 1135-1151. | MR | Zbl

[18] A.N. Tikhonov and V.Y. Arsenin, Solutions of ill-posed problems. V.H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York (1977). Translated from the Russian, Preface by translation editor Fritz John, Scripta Series in Mathematics. | MR | Zbl

[19] J.M. Varah, On the numerical solution of ill-conditioned linear systems with applications to ill-posed problems. SIAM J. Numer. Anal. 10 (1973) 257-267. Collection of articles dedicated to the memory of George E. Forsythe. | MR | Zbl

[20] J.M. Varah, A practical examination of some numerical methods for linear discrete ill-posed problems. SIAM Rev. 21 (1979) 100-111. | MR | Zbl

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