Mathematical Foundation of Interpretable Equivariant Surrogate Models

TL;DR

The paper tackles the explainability challenge for group-equivariant operators by embedding explanations into an observer-centered, diagrammatic formalism. It defines perception spaces, GEOs, GENEOs, and a surrogate distance $h_{}$ based on crossed translation pairs, plus a diagram-based complexity measure $_{}$. Learning surrogate GEOs reduces to minimizing $h_{}$ on data, enabling faithful explanations that balance interpretability with accuracy; experiments on MNIST show GEO-based surrogates can achieve competitive fidelity with lower observer-defined complexity relative to standard CNNs or MLPs. The framework generalizes to different observers and perception spaces via a monoidal-categorical formalism, offering a principled path to interpretable, symmetry-aware AI and guiding future work in formalizing XAI across domains.

Abstract

This paper introduces a rigorous mathematical framework for neural network explainability, and more broadly for the explainability of equivariant operators called Group Equivariant Operators (GEOs) based on Group Equivariant Non-Expansive Operators (GENEOs) transformations. The central concept involves quantifying the distance between GEOs by measuring the non-commutativity of specific diagrams. Additionally, the paper proposes a definition of interpretability of GEOs according to a complexity measure that can be defined according to each user preferences. Moreover, we explore the formal properties of this framework and show how it can be applied in classical machine learning scenarios, like image classification with convolutional neural networks.

Figures (1)

• Figure 1: Representation of interpretable surrogate models exemplified on MNIST.

Theorems & Definitions (1)

• definition thmcounterdefinition