Hostname: page-component-74656f9884-c52kr Total loading time: 0 Render date: 2026-11-08T00:29:45.805Z Has data issue: false hasContentIssue false
  • English
  • Français

Adaptive hard-thresholding for linear inverse problems

Published online by Cambridge University Press:  03 June 2013

Paul Rochet
Paul Rochet*
Affiliation:
Institut de Mathématiques de Toulouse, Université Paul Sabatier Toulouse III, 118 route de Narbonne, 31062 Toulouse, France. rochet@math.univ-toulouse.fr

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Keywords

Inverse problemssingular value decompositionhard-thresholding

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 485 - 499
Copyright
© EDP Sciences, SMAI, 2013

References

Abramovich, F. and Silverman, B.W., Wavelet decomposition approaches to statistical inverse problems. Biometrika 85 (1998) 115–129. Google Scholar
Bissantz, N., Hohage, T., Munk, A. and Ruymgaart, F., Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM J. Numer. Anal. 45 (2007) 2610–2636 (electronic). Google Scholar
Cavalier, L., Nonparametric statistical inverse problems. Inverse Problems 24 (2008) 034004. Google Scholar
Cavalier, L. and Golubev, G.K., Risk hull method and regularization by projections of ill-posed inverse problems. Ann. Statist. 34 (2006) 1653–1677. Google Scholar
Cavalier, L. and Hengartner, N.W., Adaptive estimation for inverse problems with noisy operators. Inverse Problems 21 (2005) 1345–1361. Google Scholar
Cavalier, L., Golubev, G.K., Picard, D. and Tsybakov, A.B., Oracle inequalities for inverse problems. Ann. Statist. 30 (2000) 843–874. Google Scholar
Efromovich, S. and Koltchinskii, V., On inverse problems with unknown operators. IEEE Trans. Inform. Theory 47 (2001) 2876–2894. Google Scholar
H.W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems, Math. Appl., vol. 375. Kluwer Academic Publishers Group, Dordrecht (1996).
Hansen, P.C., The truncated SVD as a method for regularization. BIT 27 (1987) 534–553. Google Scholar
Hansen, P.C. and O’Leary, D.P.. The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14 (1993) 1487–1503. Google Scholar
Hoffmann, M. and Reiss, M., Nonlinear estimation for linear inverse problems with error in the operator. Ann. Statist. 36 (2008) 310–336. Google Scholar
Loubes, J.M., l1 penalty for ill-posed inverse problems. Comm. Statist. Theory Methods 37 (2008) 1399–1411. Google Scholar
Loubes, J.M. and Ludeña, C., Adaptive complexity regularization for linear inverse problems. Electron. J. Stat. 2 (2008) 661–677. Google Scholar
Loubes, J.M. and Ludeña, C., Penalized estimators for non linear inverse problems. ESAIM: PS 14 (2010) 173–191. Google Scholar
F. Natterer, The mathematics of computerized tomography, Class. Appl. Math., vol. 32. SIAM, Philadelphia, PA (2001). Reprint of the 1986 original.
Scales, J.A. and Gersztenkorn, A., Robust methods in inverse theory. Inverse Problems 4 (1988) 1071–1091. Google Scholar
Stein, C.M., Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 (1981) 1135–1151. Google Scholar
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.
Varah, J.M., 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. Google Scholar
Varah, J.M., A practical examination of some numerical methods for linear discrete ill-posed problems. SIAM Rev. 21 (1979) 100–111. Google Scholar