Title: Empirical Risk Minimization as Parameter Choice Rule for General Linear Regularization Methods,
Abstract: We consider the statistical inverse problem to recover $f$ from noisy
measurements $Y = Tf + \sigma \xi$ where $\xi$ is Gaussian white noise and $T$
a compact operator between Hilbert spaces. Considering general reconstruction
methods of the form $\hat f_\alpha = q_\alpha \left(T^*T\right)T^*Y$ with an
ordered filter $q_\alpha$, we investigate the choice of the regularization
parameter $\alpha$ by minimizing an unbiased estimate of the predictive risk
$\mathbb E\left[\Vert Tf - T\hat f_\alpha\Vert^2\right]$. The corresponding
parameter $\alpha_{\mathrm{pred}}$ and its usage are well-known in the
literature, but oracle inequalities and optimality results in this general
setting are unknown. We prove a (generalized) oracle inequality, which relates
the direct risk $\mathbb E\left[\Vert f - \hat
f_{\alpha_{\mathrm{pred}}}\Vert^2\right]$ with the oracle prediction risk
$\inf_{\alpha>0}\mathbb E\left[\Vert Tf - T\hat f_{\alpha}\Vert^2\right]$. From
this oracle inequality we are then able to conclude that the investigated
parameter choice rule is of optimal order.
Finally we also present numerical simulations, which support the order
optimality of the method and the quality of the parameter choice in finite
sample situations.