  The completeness axiom renders the explanation of a post-hoc XAI method only
locally faithful to the model, i.e. for a single decision. For the trustworthy
application of XAI, in particular for high-stake decisions, a more global model
understanding is required. Recently, concept-based methods have been proposed,
which are however not guaranteed to be bound to the actual model reasoning. To
circumvent this problem, we propose Multi-dimensional Concept Discovery (MCD)
as an extension of previous approaches that fulfills a completeness relation on
the level of concepts. Our method starts from general linear subspaces as
concepts and does neither require reinforcing concept interpretability nor
re-training of model parts. We propose sparse subspace clustering to discover
improved concepts and fully leverage the potential of multi-dimensional
subspaces. MCD offers two complementary analysis tools for concepts in input
space: (1) concept activation maps, that show where a concept is expressed
within a sample, allowing for concept characterization through prototypical
samples, and (2) concept relevance heatmaps, that decompose the model decision
into concept contributions. Both tools together enable a detailed understanding
of the model reasoning, which is guaranteed to relate to the model via a
completeness relation. This paves the way towards more trustworthy
concept-based XAI. We empirically demonstrate the superiority of MCD against
more constrained concept definitions.
