This paper presents an approach to learning (deep) $n$D features equivariant under orthogonal transformations, utilizing hyperspheres and regular $n$-simplexes. Our main contributions are theoretical and tackle major challenges in geometric deep learning such as equivariance and invariance under geometric transformations. Namely, we enrich the recently developed theory of steerable 3D spherical neurons -- SO(3)-equivariant filter banks based on neurons with spherical decision surfaces -- by extending said neurons to $n$D, which we call deep equivariant hyperspheres, and enabling their multi-layer construction. 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 benchmark datasets in all but one case, additionally demonstrating a better speed/performance trade-off in all but one other case.

翻译：本文提出了一种利用超球面和正则n-单纯形学习在正交变换下等变的(深度)$n$维特征的方法。我们的主要贡献是理论性的，旨在解决几何深度学习中的主要挑战，如几何变换下的等变性和不变性。具体而言，我们将近期发展的可操控3D球形神经元理论——基于球形决策面的SO(3)-等变滤波器组——扩展至$n$维，称之为深度等变超球面，并实现其多层构造。通过在$n$维合成数据和真实世界数据上的实验，我们验证了理论贡献，并发现除一个案例外，我们的方法在基准数据集上优于竞争方法；此外，除另一个案例外，该方法还展示了更好的速度与性能权衡。