  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 at
https://github.com/pavlo-melnyk/equivariant-hyperspheres.
