  Explanation methods promise to make black-box classifiers more transparent.
As a result, it is hoped that they can act as proof for a sensible, fair and
trustworthy decision-making process of the algorithm and thereby increase its
acceptance by the end-users. In this paper, we show both theoretically and
experimentally that these hopes are presently unfounded. Specifically, we show
that, for any classifier $g$, one can always construct another classifier
$\tilde{g}$ which has the same behavior on the data (same train, validation,
and test error) but has arbitrarily manipulated explanation maps. We derive
this statement theoretically using differential geometry and demonstrate it
experimentally for various explanation methods, architectures, and datasets.
Motivated by our theoretical insights, we then propose a modification of
existing explanation methods which makes them significantly more robust.
