Nonlinear estimation for linear inverse problems with error in the operator
By Marc Hoffmann, Markus Reiß. 2008
107 citations recorded by the source corpus.
We study two nonlinear methods for statistical linear inverse problems when the operator is not known. The two constructions combine Galerkin regularization and wavelet thresholding. Their performances depend on the underlying structure of the operator, quantified by an index of sparsity. We prove their rate-optimality and adaptivity properties over Besov classes.
Subfields: Computational Mechanics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/fcfd2b9b8464fbf6edd930a9b028f5499327d478
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Computational Mechanics
- Laurent Cavalier
- Esther Klann
- Xiaohong Chen
- Demian Pouzo
- Ronny Ramlau
- Peter Maass
- Kolyan Ray
- Mark A. Lukas
In Mathematical Physics
- Laurent Cavalier
- Esther Klann
- Xiaohong Chen
- Demian Pouzo
- Ronny Ramlau
- Peter Maass
- Arnak S. Dalalyan
- Kolyan Ray
In Statistics and Probability
- Demian Pouzo
- Xiaohong Chen
- Laurent Cavalier
- Esther Klann
- Ronny Ramlau
- Peter Maass
- Andrés Santos
- Victor Chernozhukov
Citations in the corpus
- Nonparametric statistical inverse problems (2008, methodology): “This very important topic has been the subject of some recent works [13, 14, 35, 48].”
- Sparse recovery under matrix uncertainty (2008, unknown): “The Annals of Statistics 2010, Vol. 38, No. 5, 2620–2651 DOI: 10.1214/10-AOS793 © Institute of Mathematical Statistics, 2010”
- Two-step regularization methods for linear inverse problems (2006, methodology): “A first analysis of shrinkage methods for solving inverse problems was presented in [5,16].”
- Estimation of nonparametric conditional moment models with possibly nonsmooth moments (2008, result): “6The rates also coincide with those in Efromovich and Koltchinskii (2001) and Hoffmann and Reiss (2008) for statistical linear ill-posed inverse problems with unknown operators, and in Cavalier et al. (2002) and the references therein for known operators.”