Adaptive hard-thresholding for linear inverse problems

Paul Rochet

ESAIM: Probability and Statistics (2013)

  • Volume: 17, page 485-499
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Rochet, Paul. "Adaptive hard-thresholding for linear inverse problems." ESAIM: Probability and Statistics 17 (2013): 485-499. <http://eudml.org/doc/273626>.

@article{Rochet2013,
abstract = {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.},
author = {Rochet, Paul},
journal = {ESAIM: Probability and Statistics},
keywords = {inverse problems; singular value decomposition; hard-thresholding},
language = {eng},
pages = {485-499},
publisher = {EDP-Sciences},
title = {Adaptive hard-thresholding for linear inverse problems},
url = {http://eudml.org/doc/273626},
volume = {17},
year = {2013},
}

TY - JOUR
AU - Rochet, Paul
TI - Adaptive hard-thresholding for linear inverse problems
JO - ESAIM: Probability and Statistics
PY - 2013
PB - EDP-Sciences
VL - 17
SP - 485
EP - 499
AB - 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.
LA - eng
KW - inverse problems; singular value decomposition; hard-thresholding
UR - http://eudml.org/doc/273626
ER -

References

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