Risk hull method and regularization by projections of ill-posed inverse problems
By Laurent Cavalier, Yu. Golubev. 2006
86 citations recorded by the source corpus.
We study a standard method of regularization by projections of the linear inverse problem Y = Af + ∈, where ∈ is a white Gaussian noise, and A is a known compact operator with singular values converging to zero with polynomial decay. The unknown function f is recovered by a projection method using the singular value decomposition of A. The bandwidth choice of this projection regularization is governed by a data-driven procedure which is based on the principle of risk hull minimization. We provide nonasymptotic upper bounds for the mean square risk of this method and we show, in particular, that in numerical simulations this approach may substantially improve the classical method of unbiased risk estimation.
Subfields: Applied Mathematics, Computational Mechanics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/5e42c39531f211bd7768359084044296dbab299c
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In Applied Mathematics
- Mark A. Lukas
- Frank Bauer
- A. van der Vaart
- J. H. Zanten
- Botond Szab'o
- B. Knapik
- Laurent Cavalier
- Lorenzo Rosasco
In Computational Mechanics
- Mark A. Lukas
- Frank Bauer
- B. Knapik
- A. van der Vaart
- J. H. Zanten
- Botond Szab'o
- Laurent Cavalier
- Kolyan Ray
In Statistics and Probability
- Botond Szab'o
- B. Knapik
- A. van der Vaart
- J. H. Zanten
- Kolyan Ray
- Frank Bauer
- Mark A. Lukas
- Judith Rousseau
Citations in the corpus
- Comparing parameter choice methods for regularization of ill-posed problems (2011, methodology): “– unbiased risk method [27, 100]: behaves like the modified discrepancy principle with a particular tuning constant”
- Nonparametric statistical inverse problems (2008, methodology): “The following oracle inequality (whose proof can be found in [10]) provides an upper bound for the mean square risk of the RHM approach.”
- Bayes procedures for adaptive inference in inverse problems for the white noise model (2012, methodology): “another approach to adaptation in inverse problems is the risk hull method studied in [11].”