Nonlinear estimation for linear inverse problems with error in the operator

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.

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)

References in corpus (2)

• Prediction in functional linear regression
• Nonparametric methods for inference in the presence of instrumental variables

Cited by in corpus (9)

• Thresholding Projection Estimators in Functional Linear Models
• Adaptive functional linear regression
• A deconvolution approach to estimation of a common shape in a shifted curves model
• Adaptive Gaussian inverse regression with partially unknown operator
• Variational Multiscale Nonparametric Regression: Smooth Functions
• Bayesian inverse problems with unknown operators
• Spatially inhomogeneous linear inverse problems with possible singularities
• Bayesian Inverse Problems with Heterogeneous Variance
• Deep learning for inverse problems with unknown operator