The Shape of Explanations: A Topological Account of Rule-Based Explanations in Machine Learning

Rule-based explanations provide simple reasons explaining the behavior of machine learning classifiers at given points in the feature space. Several recent methods (Anchors, LORE, etc.) purport to generate rule-based explanations for arbitrary or black-box classifiers. But what makes these methods work in general? We introduce a topological framework for rule-based explanation methods and provide a characterization of explainability in terms of the definability of a classifier relative to an explanation scheme. We employ this framework to consider various explanation schemes and argue that the preferred scheme depends on how much the user knows about the domain and the probability measure over the feature space.