Nonlinear estimation for linear inverse problems with error in the operator

2008/02/01 by Marc Hoffmann, Markus Reiss · 1 citation
Engineering · Mathematics · #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques #Statistical Methods and Inference #math.ST #msc:62G07 #msc:65J20 #stat.TH

paper · pdf · doi:10.1214/009053607000000721

published as Annals of Statistics 2008, Vol. 36, No. 1, 310-336 · Published in at http://dx.doi.org/10.1214/009053607000000721 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

openalex publication_date 2008/02/01 · arxiv created 2008/03/13 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/01

Abstract

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.

Citations

• Nonparametric methods for inference in the presence of instrumental variables

Cited by

• Estimation and inference in error-in-operator model