  Interpretability methods are valuable only if their explanations faithfully
describe the explained model. In this work, we consider neural networks whose
predictions are invariant under a specific symmetry group. This includes
popular architectures, ranging from convolutional to graph neural networks. Any
explanation that faithfully explains this type of model needs to be in
agreement with this invariance property. We formalize this intuition through
the notion of explanation invariance and equivariance by leveraging the
formalism from geometric deep learning. Through this rigorous formalism, we
derive (1) two metrics to measure the robustness of any interpretability method
with respect to the model symmetry group; (2) theoretical robustness guarantees
for some popular interpretability methods and (3) a systematic approach to
increase the invariance of any interpretability method with respect to a
symmetry group. By empirically measuring our metrics for explanations of models
associated with various modalities and symmetry groups, we derive a set of 5
guidelines to allow users and developers of interpretability methods to produce
robust explanations.
