$α$-TCAV: A Unified Framework for Testing with Concept Activation Vectors

Concept Activation Vectors (CAV) and the Testing with CAVs (TCAV) method are fundamental tools for concept-based explainability in deep learning, yet their practical utility is limited by the variability across different samples of concept and non-concept examples. We analyze this stochastic nature of CAVs, and identify a fundamental flaw in the standard TCAV score, its instability by using a discontinuous indicator function. To address this, we introduce α-TCAV, a parameterized unified framework that replaces the discontinuous indicator with a smooth function, which recovers TCAV as a limiting case. Under realistic assumptions, we characterize the score distributions and derive principled parameter choices: one matches the mean of its averaged counterpart, Multi-TCAV, with no larger variance using a single CAV; the other provides a calibrated probability that the concept positively influences the model output. Our analysis and experiments provide practical guidance and challenge the established practice of splitting the sampling budget across multiple CAVs.