# A topological model for partial equivariance in deep learning and data analysis

arXiv: https://arxiv.org/abs/2308.13357 · HTML: https://mlsanity.com/paper/2308.13357/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 26 references that are in the map.
- First posted on arXiv 25 Aug 2023.

## Builds on (most cited first, 14 of 14)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Learning Partial Equivariances from Data](https://mlsanity.com/paper/2110.10211/) (2021)
- [Relaxing Equivariance Constraints with Non-stationary Continuous Filters](https://mlsanity.com/paper/2204.07178/) (2022)
- [Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning](https://mlsanity.com/paper/1812.11832/) (2018)
- [On the finite representation of group equivariant operators via permutant measures](https://mlsanity.com/paper/2008.06340/) (2020)
- [GENEOnet: A new machine learning paradigm based on Group Equivariant Non-Expansive Operators. An application to protein pocket detection](https://mlsanity.com/paper/2202.00451/) (2022)
- [On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operators](https://mlsanity.com/paper/2103.02543/) (2021)
- [A probabilistic result on impulsive noise reduction in Topological Data Analysis through Group Equivariant Non-Expansive Operators](https://mlsanity.com/paper/2202.14021/) (2022)

## Built on by (most cited first, 1 of 1)

- [GENEOnet: Statistical analysis supporting explainability and trustworthiness](https://mlsanity.com/paper/2503.09199/) (2025)

## Abstract (arXiv)

In this article, we propose a topological model to encode partial equivariance in neural networks. To this end, we introduce a class of operators, called P-GENEOs, that change data expressed by measurements, respecting the action of certain sets of transformations, in a non-expansive way. If the set of transformations acting is a group, then we obtain the so-called GENEOs. We then study the spaces of measurements, whose domains are subject to the action of certain self-maps, and the space of P-GENEOs between these spaces. We define pseudo-metrics on them and show some properties of the resulting spaces. In particular, we show how such spaces have convenient approximation and convexity properties.