  Hallucinations are an inescapable consequence of solving inverse problems
with deep neural networks. The expressiveness of recent generative models is
the reason why they can yield results far superior to conventional
regularizers; it can also lead to realistic-looking but incorrect features,
potentially undermining the trust in important aspects of the reconstruction.
We present a practical and computationally efficient method to determine, which
regions in the solutions of inverse problems with data-driven priors are prone
to hallucinations. By computing the diagonal elements of the Fisher information
matrix of the likelihood and the data-driven prior separately, we can flag
regions where the information is prior-dominated. Our diagnostic can directly
be compared to the reconstructed solutions and enables users to decide if
measurements in such regions are robust for their application. Our method
scales linearly with the number of parameters and is thus applicable in
high-dimensional settings, allowing it to be rolled out broadly for the
large-volume data products of future wide-field surveys.
