  Concept-based explanations permit to understand the predictions of a deep
neural network (DNN) through the lens of concepts specified by users. Existing
methods assume that the examples illustrating a concept are mapped in a fixed
direction of the DNN's latent space. When this holds true, the concept can be
represented by a concept activation vector (CAV) pointing in that direction. In
this work, we propose to relax this assumption by allowing concept examples to
be scattered across different clusters in the DNN's latent space. Each concept
is then represented by a region of the DNN's latent space that includes these
clusters and that we call concept activation region (CAR). To formalize this
idea, we introduce an extension of the CAV formalism that is based on the
kernel trick and support vector classifiers. This CAR formalism yields global
concept-based explanations and local concept-based feature importance. We prove
that CAR explanations built with radial kernels are invariant under latent
space isometries. In this way, CAR assigns the same explanations to latent
spaces that have the same geometry. We further demonstrate empirically that
CARs offer (1) more accurate descriptions of how concepts are scattered in the
DNN's latent space; (2) global explanations that are closer to human concept
annotations and (3) concept-based feature importance that meaningfully relate
concepts with each other. Finally, we use CARs to show that DNNs can
autonomously rediscover known scientific concepts, such as the prostate cancer
grading system.
