A number of regularization methods for discrete inverse problems consist in
considering weighted versions of the usual least square solution. However,
these so-called filter methods are generally restricted to monotonic
transformations, e.g. the Tikhonov regularization or the spectral cut-off. In
this paper, we point out that in several cases, non-monotonic sequences of
filters are more efficient. We study a regularization method that naturally
extends the spectral cut-off procedure to non-monotonic sequences and provide
several oracle inequalities, showing the method to be nearly optimal under mild
assumptions. Then, we extend the method to inverse problems with noisy operator
and provide efficiency results in a newly introduced conditional framework
