# On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operators

arXiv: https://arxiv.org/abs/2103.02543 · HTML: https://mlsanity.com/paper/2103.02543/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 26 references that are in the map.
- First posted on arXiv 3 Mar 2021.

## Builds on (most cited first, 9 of 9)

- [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)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Understanding Deep Convolutional Networks](https://mlsanity.com/paper/1601.04920/) (2016)
- [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 Invariance and Selectivity in Representation Learning](https://mlsanity.com/paper/1503.05938/) (2015)
- [On the finite representation of group equivariant operators via permutant measures](https://mlsanity.com/paper/2008.06340/) (2020)
- [Combining persistent homology and invariance groups for shape comparison](https://mlsanity.com/paper/1312.7219/) (2013)
- [Position paper: Towards an observer-oriented theory of shape comparison](https://mlsanity.com/paper/1603.02008/) (2016)

## Built on by (most cited first, 3 of 3)

- [Mathematical Foundation of Interpretable Equivariant Surrogate Models](https://mlsanity.com/paper/2503.01942/) (2025)
- [Low-Resource White-Box Semantic Segmentation of Supporting Towers on 3D Point Clouds via Signature Shape Identification](https://mlsanity.com/paper/2306.07809/) (2023)
- [A topological model for partial equivariance in deep learning and data analysis](https://mlsanity.com/paper/2308.13357/) (2023)

## Abstract (arXiv)

Group equivariant non-expansive operators have been recently proposed as basic components in topological data analysis and deep learning. In this paper we study some geometric properties of the spaces of group equivariant operators and show how a space F of group equivariant non-expansive operators can be endowed with the structure of a Riemannian manifold, so making available the use of gradient descent methods for the minimization of cost functions on F. As an application of this approach, we also describe a procedure to select a finite set of representative group equivariant non-expansive operators in the considered manifold.