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 baselines for small training data sets in all but one case.

翻译：本文提出了一种利用超球面和正则$n$单纯形学习正交变换下（深度）$n$维等变特征的方法。我们的主要贡献是理论性的，旨在解决几何深度学习中的关键挑战，如几何变换下的等变性和不变性。具体而言，我们通过将最近发展的可操控3D球形神经元理论——基于具有球形决策面的神经元的SO(3)等变滤波器组——推广到$n$维（称之为深度等变超球面），并实现其多层构建，从而丰富了该理论。利用$n$维合成数据和真实世界数据，我们通过实验验证了理论贡献，并发现除一种情况外，我们的方法在小训练数据集上的表现均优于基线方法。