Probabilistic Lipschitzness and the Stable Rank for Comparing Explanation Models

Explainability models are now prevalent within machine learning to address the black-box nature of neural networks. The question now is which explainability model is most effective. Probabilistic Lipschitzness has demonstrated that the smoothness of a neural network is fundamentally linked to the quality of post hoc explanations. In this work, we prove theoretical lower bounds on the probabilistic Lipschitzness of Integrated Gradients, LIME and SmoothGrad. We propose a novel metric using probabilistic Lipschitzness, normalised astuteness, to compare the robustness of explainability models. Further, we prove a link between the local Lipschitz constant of a neural network and its stable rank. We then demonstrate that the stable rank of a neural network provides a heuristic for the robustness of explainability models.