Oracle inequalities for inverse problems

We consider a sequence space model of statistical linear inverse problems where we need to estimate a function $f$ from indirect noisy observations. Let a finite set $\Lambda$ of linear estimators be given. Our aim is to mimic the estimator in $\Lambda$ that has the smallest risk on the true $f$. Under general conditions, we show that this can be achieved by simple minimization of an unbiased risk estimator, provided the singular values of the operator of the inverse problem decrease as a power law. The main result is a nonasymptotic oracle inequality that is shown to be asymptotically exact. This inequality can also be used to obtain sharp minimax adaptive results. In particular, we apply it to show that minimax adaptation on ellipsoids in the multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal nonadaptive procedures.