# Mathematical Foundation of Interpretable Equivariant Surrogate Models

arXiv: https://arxiv.org/abs/2503.01942 · HTML: https://mlsanity.com/paper/2503.01942/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 42 references that are in the map.
- First posted on arXiv 3 Mar 2025.

## Builds on (most cited first, 22 of 22)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [A Survey Of Methods For Explaining Black Box Models](https://mlsanity.com/paper/1802.01933/) (2018)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [From Anecdotal Evidence to Quantitative Evaluation Methods: A Systematic Review on Evaluating Explainable AI](https://mlsanity.com/paper/2201.08164/) (2022)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Benchmarking and Survey of Explanation Methods for Black Box Models](https://mlsanity.com/paper/2102.13076/) (2021)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [Logic Explained Networks](https://mlsanity.com/paper/2108.05149/) (2021)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [Interpretable and Differentially Private Predictions](https://mlsanity.com/paper/1906.02004/) (2019)
- [Explain, Edit, and Understand: Rethinking User Study Design for Evaluating Model Explanations](https://mlsanity.com/paper/2112.09669/) (2021)
- [Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning](https://mlsanity.com/paper/1812.11832/) (2018)
- [Interpretability is in the Mind of the Beholder: A Causal Framework for Human-interpretable Representation Learning](https://mlsanity.com/paper/2309.07742/) (2023)
- [On Invariance and Selectivity in Representation Learning](https://mlsanity.com/paper/1503.05938/) (2015)
- [XAI Handbook: Towards a Unified Framework for Explainable AI](https://mlsanity.com/paper/2105.06677/) (2021)
- [Categorical Foundations of Explainable AI: A Unifying Theory](https://mlsanity.com/paper/2304.14094/) (2023)
- [Combining persistent homology and invariance groups for shape comparison](https://mlsanity.com/paper/1312.7219/) (2013)
- [Generalized Permutants and Graph GENEOs](https://mlsanity.com/paper/2206.14798/) (2022)
- [A novel approach to graph distinction through GENEOs and permutants](https://mlsanity.com/paper/2406.08045/) (2024)
- [On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operators](https://mlsanity.com/paper/2103.02543/) (2021)
- [Exploration-oriented sampling strategies for global surrogate modeling: A comparison between one-stage and adaptive methods](https://mlsanity.com/paper/2211.08534/) (2022)

## Built on by (most cited first, 5 of 5)

- [Actionable Interpretability Must Be Defined in Terms of Symmetries](https://mlsanity.com/paper/2601.12913/) (2026)
- [Mixture of Concept Bottleneck Experts](https://mlsanity.com/paper/2602.02886/) (2026)
- [Logic Explanation of AI Classifiers by Categorical Explaining Functors](https://mlsanity.com/paper/2503.16203/) (2025)
- [Reconstruction of SINR Maps from Sparse Measurements using Group Equivariant Non-Expansive Operators](https://mlsanity.com/paper/2507.19349/) (2025)
- [An Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learning](https://mlsanity.com/paper/2601.03910/) (2026)

## Abstract (arXiv)

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.