Risk hull method for spectral regularization in linear statistical inverse problems

Abstract

Abstract. We consider in this paper the statistical linear inverse problem Y = Af + ξ where A denotes a compact operator, a noise level and ξ a stochastic noise. The unknown function f has to be recovered from the indirect measurement Y . We are interested in the following approach: given a family of estimators, we want to select the best possible one. In this context, the unbiased risk estimation (URE) method is rather popular. Nevertheless, it is also very unstable. Recently, Cavalier and Golubev (2006) introduced the risk hull minimization (RHM) method. It significantly improves the performances of the standard URE procedure. However, it only concerns projection rules. Using recent developments on ordered processes, we prove in this paper that it can be extended to a large class of linear estimators.Mathematics Subject Classification. 62G05, 62G20.

Bayes procedures for adaptive inference in inverse problems for the white noise model

“…In this paper the authors consider spectral cut-off estimators and provide oracle inequalities. An extension of their approach is presented in [25]. The link between the penalized blockwise Stein's rule and the risk hull method is presented in [26].…”