Risk hull method and regularization by projections of ill-posed inverse problems

Abstract

We study a standard method of regularization by projections of the linear inverse problem $Y=Af+\epsilon$, where $\epsilon$ 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.

Published at http://dx.doi.org/10.1214/009053606000000542 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (17)

• Beyond the Bakushinskii veto: Regularising linear inverse problems without knowing the noise distribution
• Uniform bounds for norms of sums of independent random functions
• Adaptive complexity regularization for linear inverse problems
• On universal oracle inequalities related to high-dimensional linear models
• Empirical Risk Minimization as Parameter Choice Rule for General Linear Regularization Methods
• Bayes procedures for adaptive inference in inverse problems for the white noise model
• Minimax Goodness-of-Fit Testing in Ill-Posed Inverse Problems with Partially Unknown Operators
• Bootstrap tuning in ordered model selection
• Spatially inhomogeneous linear inverse problems with possible singularities
• Upper functions for positive random functionals
• Optimal regularized hypothesis testing in statistical inverse problems
• Upper functions for $L_p$-norm of gaussian random fields
• Adaptation in a class of linear inverse problems
• A Threshold Regularization Method for Inverse Problems
• Estimation in the convolution structure density model. Part I: oracle inequalities
• Model selection in a sparse heterogeneous framework
• Adaptive spectral regularizations of high dimensional linear models