# Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning

arXiv: https://arxiv.org/abs/1812.11832 · HTML: https://mlsanity.com/paper/1812.11832/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Dec 2018.

## Built on by (most cited first, 24 of 24)

- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Model-centric Data Manifold: the Data Through the Eyes of the Model](https://mlsanity.com/paper/2104.13289/) (2021)
- [Generalized Permutants and Graph GENEOs](https://mlsanity.com/paper/2206.14798/) (2022)
- [Mathematical Foundation of Interpretable Equivariant Surrogate Models](https://mlsanity.com/paper/2503.01942/) (2025)
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- [Achieving Constraints in Neural Networks: A Stochastic Augmented Lagrangian Approach](https://mlsanity.com/paper/2310.16647/) (2023)
- [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)
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- [A topological model for partial equivariance in deep learning and data analysis](https://mlsanity.com/paper/2308.13357/) (2023)
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- [Leveraging The Topological Consistencies of Learning in Deep Neural Networks](https://mlsanity.com/paper/2111.15651/) (2021)
- [Persistence-based operators in machine learning](https://mlsanity.com/paper/2212.13985/) (2022)
- [A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams](https://mlsanity.com/paper/2402.14489/) (2024)
- [GENEOnet: Statistical analysis supporting explainability and trustworthiness](https://mlsanity.com/paper/2503.09199/) (2025)
- [Reconstruction of SINR Maps from Sparse Measurements using Group Equivariant Non-Expansive Operators](https://mlsanity.com/paper/2507.19349/) (2025)
- [Hierarchical topological clustering](https://mlsanity.com/paper/2601.00892/) (2025)
- [An Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learning](https://mlsanity.com/paper/2601.03910/) (2026)
- [G-LoG Bi-filtration for Medical Image Classification](https://mlsanity.com/paper/2602.18329/) (2026)

## Abstract (arXiv)

The aim of this paper is to provide a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. We define group-equivariant non-expansive operators (GENEOs), which are maps between function spaces associated with groups of transformations. We study the topological and metric properties of the space of GENEOs to evaluate their approximating power and set the basis for general strategies to initialise and compose operators. We begin by defining suitable pseudo-metrics for the function spaces, the equivariance groups, and the set of non-expansive operators. Basing on these pseudo-metrics, we prove that the space of GENEOs is compact and convex, under the assumption that the function spaces are compact and convex. These results provide fundamental guarantees in a machine learning perspective. We show examples on the MNIST and fashion-MNIST datasets. By considering isometry-equivariant non-expansive operators, we describe a simple strategy to select and sample operators, and show how the selected and sampled operators can be used to perform both classical metric learning and an effective initialisation of the kernels of a convolutional neural network.