Recent interpretability methods propose using concept-based explanations to translate the internal representations of deep learning models into a language that humans are familiar with: concepts. This requires understanding which concepts are present in the representation space of a neural network. One popular method for finding concepts is Concept Activation Vectors (CAVs), which are learnt using a probe dataset of concept exemplars. In this work, we investigate three properties of CAVs. CAVs may be: (1) inconsistent between layers, (2) entangled with different concepts, and (3) spatially dependent. Each property provides both challenges and opportunities in interpreting models. We introduce tools designed to detect the presence of these properties, provide insight into how they affect the derived explanations, and provide recommendations to minimise their impact. Understanding these properties can be used to our advantage. For example, we introduce spatially dependent CAVs to test if a model is translation invariant with respect to a specific concept and class. Our experiments are performed on ImageNet and a new synthetic dataset, Elements. Elements is designed to capture a known ground truth relationship between concepts and classes. We release this dataset to facilitate further research in understanding and evaluating interpretability methods.

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