O$n$ Learning Deep O($n$)-Equivariant Hyperspheres
Pavlo Melnyk, Michael Felsberg, Mårten Wadenbäck, Andreas Robinson, Cuong Le
Abstract
In this paper, we utilize hyperspheres and regular $n$-simplexes and propose an approach to learning deep features equivariant under the transformations of $n$D 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 $n$D, 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 on [GitHub](https://github.com/pavlo-melnyk/equivariant-hyperspheres).
BibTeX
@inproceedings{
melnyk2024on,
title={O\$n\$ Learning Deep O(\$n\$)-Equivariant Hyperspheres},
author={Pavlo Melnyk and Michael Felsberg and M{\r{a}}rten Wadenb{\"a}ck and Andreas Robinson and Cuong Le},
booktitle={Forty-first International Conference on Machine Learning},
year={2024},
url={https://openreview.net/forum?id=pFWmHUdJE5}
}