Risk hull method for spectral regularization in linear statistical inverse problems

Clément Marteau

ESAIM: Probability and Statistics (2010)

  • Volume: 14, page 409-434
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Marteau, Clément. "Risk hull method for spectral regularization in linear statistical inverse problems." ESAIM: Probability and Statistics 14 (2010): 409-434. <http://eudml.org/doc/250852>.

@article{Marteau2010,
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. },
author = {Marteau, Clément},
journal = {ESAIM: Probability and Statistics},
keywords = {Inverse problems; oracle inequality; ordered process; risk hull and Tikhonov estimation; inverse problems},
language = {eng},
month = {12},
pages = {409-434},
publisher = {EDP Sciences},
title = {Risk hull method for spectral regularization in linear statistical inverse problems},
url = {http://eudml.org/doc/250852},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Marteau, Clément
TI - Risk hull method for spectral regularization in linear statistical inverse problems
JO - ESAIM: Probability and Statistics
DA - 2010/12//
PB - EDP Sciences
VL - 14
SP - 409
EP - 434
AB - 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.
LA - eng
KW - Inverse problems; oracle inequality; ordered process; risk hull and Tikhonov estimation; inverse problems
UR - http://eudml.org/doc/250852
ER -

References

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