Risk hull method for spectral regularization in linear statistical inverse problems

Institut de Mathématiques, Université de Toulouse, INSA - Département GMM, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France; clement.marteau@insa-toulouse.fr

Received: 7 July 2008
Revised: 7 April 2009

Abstract

We consider in this paper the statistical linear inverse problem Y = Af + ϵξ where A denotes a compact operator, ϵ 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.