Mathematical Foundation of Interpretable Equivariant Surrogate Models

2025-03-03

AI Summary

Neural network group-equivariant operators (GEOs) lack a rigorous mathematical foundation for interpretability. Method: We propose the first axiomatic framework for GEO interpretability, introducing a noncommutativity metric grounded in group representation theory and category theory to quantify distances between GEOs; integrating user-specified complexity preferences to formally define interpretability; and leveraging nonexpansive operator theory to ensure stability and generalization. Contribution/Results: We establish the first axiomatized interpretability system for GEOs driven by Group-Equivariant Neural Equations Operators (GENEOs), transforming interpretability into measurable, tunable, and provable mathematical properties. We prove theoretical robustness under perturbations and empirically validate the framework on CNN-based image classification, demonstrating consistency and verifiability between local and global explanations. This work bridges abstract algebraic structure with practical interpretability requirements, enabling principled analysis of equivariant deep learning models.

Technology Category

• Computer Vision: Interpretability, Explainability, and Transparency
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Philosophy and Ethics of AI: Accountability, Interpretability & Explainability

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• Develops a mathematical framework for neural network explainability.
• Quantifies distance between Group Equivariant Operators (GEOs).
• Proposes interpretability definition for GEOs based on complexity.

Innovation

• Rigorous framework for neural network explainability
• Distance quantification via non-commutativity measurement
• User-defined complexity measure for interpretability