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.

翻译：本文利用超球面和正则 $n$-单纯形，提出了一种学习深度特征的方法，该特征在 $n$ 维反射和旋转变换下具有等变性，这些变换涵盖于强大的 O$(n)$ 群之中。具体而言，我们提出了具有球形决策面的 O$(n)$-等变神经元，该神经元可推广至任意维度 $n$，我们将其称为深度等变超球面。我们展示了如何将这些神经元组合成一个直接基于输入点操作的网络，并提出了一个基于两点与球面关系的不变算子，该算子经证明为格拉姆矩阵。通过使用 $n$ 维合成数据和真实数据，我们实验验证了理论贡献，并发现我们的方法在 O$(n)$-等变基准数据集（分类和回归）上优于现有竞争方法，展现了良好的速度/性能权衡。