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

arXiv: https://arxiv.org/abs/2605.15688 · HTML: https://mlsanity.com/paper/2605.15688/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 42 of its 77 references that are in the map.
- First posted on arXiv 15 May 2026.

## Builds on (most cited first, 30 of 42)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Interpretability Beyond Feature Attribution: Quantitative Testing with Concept Activation Vectors (TCAV)](https://mlsanity.com/paper/1711.11279/) (2017)
- [Concept Bottleneck Models](https://mlsanity.com/paper/2007.04612/) (2020)
- [A systematic study of the class imbalance problem in convolutional neural networks](https://mlsanity.com/paper/1710.05381/) (2017)
- [Towards Robust Interpretability with Self-Explaining Neural Networks](https://mlsanity.com/paper/1806.07538/) (2018)
- [Post-hoc Concept Bottleneck Models](https://mlsanity.com/paper/2205.15480/) (2022)
- [Being Bayesian, Even Just a Bit, Fixes Overconfidence in ReLU Networks](https://mlsanity.com/paper/2002.10118/) (2020)
- [The Disagreement Problem in Explainable Machine Learning: A Practitioner's Perspective](https://mlsanity.com/paper/2202.01602/) (2022)
- [From Attribution Maps to Human-Understandable Explanations through Concept Relevance Propagation](https://mlsanity.com/paper/2206.03208/) (2022)
- [Acquisition of Chess Knowledge in AlphaZero](https://mlsanity.com/paper/2111.09259/) (2021)
- [A Holistic Approach to Unifying Automatic Concept Extraction and Concept Importance Estimation](https://mlsanity.com/paper/2306.07304/) (2023)
- [Concept-based Explainable Artificial Intelligence: A Survey](https://mlsanity.com/paper/2312.12936/) (2023)
- [Explaining the Explainer: A First Theoretical Analysis of LIME](https://mlsanity.com/paper/2001.03447/) (2020)
- [A Random Matrix Approach to Neural Networks](https://mlsanity.com/paper/1702.05419/) (2017)
- [Concept Activation Regions: A Generalized Framework For Concept-Based Explanations](https://mlsanity.com/paper/2209.11222/) (2022)
- [Random Matrix Theory Proves that Deep Learning Representations of GAN-data Behave as Gaussian Mixtures](https://mlsanity.com/paper/2001.08370/) (2020)
- [Concept Bottleneck Large Language Models](https://mlsanity.com/paper/2412.07992/) (2024)
- [Bridging the Human-AI Knowledge Gap: Concept Discovery and Transfer in AlphaZero](https://mlsanity.com/paper/2310.16410/) (2023)
- [Steering Large Language Models using Conceptors: Improving Addition-Based Activation Engineering](https://mlsanity.com/paper/2410.16314/) (2024)
- [Controlling Large Language Models Through Concept Activation Vectors](https://mlsanity.com/paper/2501.05764/) (2025)
- [Navigating Neural Space: Revisiting Concept Activation Vectors to Overcome Directional Divergence](https://mlsanity.com/paper/2202.03482/) (2022)
- [Semi-analytical approximations to statistical moments of sigmoid and softmax mappings of normal variables](https://mlsanity.com/paper/1703.00091/) (2017)
- [Looking Deeper into Tabular LIME](https://mlsanity.com/paper/2008.11092/) (2020)
- [From Hope to Safety: Unlearning Biases of Deep Models via Gradient Penalization in Latent Space](https://mlsanity.com/paper/2308.09437/) (2023)
- [Explaining Explainability: Recommendations for Effective Use of Concept Activation Vectors](https://mlsanity.com/paper/2404.03713/) (2024)

## Abstract (arXiv)

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.