# On Learning Deep O(n)-Equivariant Hyperspheres

arXiv: https://arxiv.org/abs/2305.15613 · HTML: https://mlsanity.com/paper/2305.15613/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 27 references that are in the map.
- First posted on arXiv 24 May 2023.

## Builds on (most cited first, 4 of 4)

- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [TetraSphere: A Neural Descriptor for O(3)-Invariant Point Cloud Analysis](https://mlsanity.com/paper/2211.14456/) (2022)

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

- [Benchmark Dataset for Catalysis on 2D MXenes](https://mlsanity.com/paper/2606.00794/) (2026)

## Abstract (arXiv)

In this paper, we utilize hyperspheres and regular n-simplexes and propose an approach to learning deep features equivariant under the transformations of nD reflections and rotations, encompassed by the powerful group of O(n). Namely, we propose O(n)-equivariant neurons with spherical decision surfaces that generalize to any dimension n, which we call Deep Equivariant Hyperspheres. We demonstrate how to combine them in a network that directly operates on the basis of the input points and propose an invariant operator based on the relation between two points and a sphere, which as we show, turns out to be a Gram matrix. Using synthetic and real-world data in nD, we experimentally verify our theoretical contributions and find that our approach is superior to the competing methods for O(n)-equivariant benchmark datasets (classification and regression), demonstrating a favorable speed/performance trade-off. The code is available at https://github.com/pavlo-melnyk/equivariant-hyperspheres.