Adaptive hard-thresholding for linear inverse problems

Institut de Mathématiques de Toulouse, Université Paul Sabatier Toulouse III, 118 route de Narbonne, 31062 Toulouse, France
rochet@math.univ-toulouse.fr

Received: 2 May 2011
Revised: 9 November 2011

Abstract

A number of regularization methods for discrete inverse problems consist in considering weighted versions of the usual least square solution. These filter methods are generally restricted to monotonic transformations, e.g. the Tikhonov regularization or the spectral cut-off. However, in several cases, non-monotonic sequences of filters may appear more appropriate. In this paper, we study a hard-thresholding regularization method that extends the spectral cut-off procedure to non-monotonic sequences. We provide several oracle inequalities, showing the method to be nearly optimal under mild assumptions. Contrary to similar methods discussed in the literature, we use here a non-linear threshold that appears to be adaptive to all degrees of irregularity, whether the problem is mildly or severely ill-posed. Finally, we extend the method to inverse problems with noisy operator and provide efficiency results in a conditional framework.