Oracle inequality for a statistical raus–gfrerer-type rule

We consider statistical linear inverse problems in Hilbert spaces. Approximate solutions are sought within a class of one-parameter linear regularization schemes in which the choice of the parameter is crucial for controlling the root mean squared error. A variant of the Raus–Gfrerer rule is proposed and analyzed. It is shown that this parameter choice gives rise to error bounds in terms of oracle inequalities, which in turn provide order optimal error bounds (up to logarithmic factors). These bounds are established only for solutions that obey certain self-similarity properties. The proof of the main result relies on some auxiliary error analysis for linear inverse problems under general noise assumptions, which might be interesting in its own right.