Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations
Loading...
Date
2017
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
This paper discusses the properties of certain risk estimators recently proposed to
choose regularization parameters in ill-posed problems. A simple approach is Stein's unbiased
risk estimator (SURE), which estimates the risk in the data space, while a recent
modification (GSURE) estimates the risk in the space of the unknown variable. It seems
intuitive that the latter is more appropriate for ill-posed problems, since the properties
in the data space do not tell much about the quality of the reconstruction. We provide
theoretical studies of both estimators for linear Tikhonov regularization in a finite
dimensional setting and estimate the quality of the risk estimators, which also leads to
asymptotic convergence results as the dimension of the problem tends to infinity. Unlike
previous papers, who studied image processing problems with a very low degree of
ill-posedness, we are interested in the behavior of the risk estimators for increasing illposedness.
Interestingly, our theoretical results indicate that the quality of the GSURE
risk can deteriorate asymptotically for ill-posed problems, which is confirmed by a detailed
numerical study. The latter shows that in many cases the GSURE estimator leads
to extremely small regularization parameters, which obviously cannot stabilize the reconstruction.
Similar but less severe issues with respect to robustness also appear for the
SURE estimator, which in comparison to the rather conservative discrepancy principle
leads to the conclusion that regularization parameter choice based on unbiased risk estimation
is not a reliable procedure for ill-posed problems. A similar numerical study for
sparsity regularization demonstrates that the same issue appears in nonlinear variational
regularization approaches.
Description
Table of contents
Keywords
ill-posed problems, discrepancy principle, Stein's method, risk estimators, regularization parameter choice