  Deploying AI-powered systems requires trustworthy models supporting effective
human interactions, going beyond raw prediction accuracy. Concept bottleneck
models promote trustworthiness by conditioning classification tasks on an
intermediate level of human-like concepts. This enables human interventions
which can correct mispredicted concepts to improve the model's performance.
However, existing concept bottleneck models are unable to find optimal
compromises between high task accuracy, robust concept-based explanations, and
effective interventions on concepts -- particularly in real-world conditions
where complete and accurate concept supervisions are scarce. To address this,
we propose Concept Embedding Models, a novel family of concept bottleneck
models which goes beyond the current accuracy-vs-interpretability trade-off by
learning interpretable high-dimensional concept representations. Our
experiments demonstrate that Concept Embedding Models (1) attain better or
competitive task accuracy w.r.t. standard neural models without concepts, (2)
provide concept representations capturing meaningful semantics including and
beyond their ground truth labels, (3) support test-time concept interventions
whose effect in test accuracy surpasses that in standard concept bottleneck
models, and (4) scale to real-world conditions where complete concept
supervisions are scarce.
