Group equivariant convolutional neural networks (G-CNNs) have been successfully applied in geometric deep learning. Typically, G-CNNs have the advantage over CNNs that they do not waste network capacity on training symmetries that should have been hard-coded in the network. The recently introduced framework of PDE-based G-CNNs (PDE-G-CNNs) generalises G-CNNs. PDE-G-CNNs have the core advantages that they simultaneously 1) reduce network complexity, 2) increase classification performance, and 3) provide geometric interpretability. Their implementations primarily consist of linear and morphological convolutions with kernels. In this paper we show that the previously suggested approximative morphological kernels do not always accurately approximate the exact kernels accurately. More specifically, depending on the spatial anisotropy of the Riemannian metric, we argue that one must resort to sub-Riemannian approximations. We solve this problem by providing a new approximative kernel that works regardless of the anisotropy. We provide new theorems with better error estimates of the approximative kernels, and prove that they all carry the same reflectional symmetries as the exact ones. We test the effectiveness of multiple approximative kernels within the PDE-G-CNN framework on two datasets, and observe an improvement with the new approximative kernels. We report that the PDE-G-CNNs again allow for a considerable reduction of network complexity while having comparable or better performance than G-CNNs and CNNs on the two datasets. Moreover, PDE-G-CNNs have the advantage of better geometric interpretability over G-CNNs, as the morphological kernels are related to association fields from neurogeometry.

翻译：群等变卷积神经网络（G-CNNs）已成功应用于几何深度学习。通常，G-CNNs相比CNNs的优势在于它们不会将网络容量浪费在本应硬编码到网络中的对称性训练上。最近提出的基于偏微分方程的G-CNNs（PDE-G-CNNs）框架推广了G-CNNs。PDE-G-CNNs的核心优势在于它们能同时：1）降低网络复杂度，2）提升分类性能，以及3）提供几何可解释性。其实现主要依赖于带核的线性卷积和形态学卷积。本文表明，先前建议的近似形态学核并非总能准确逼近精确核。具体而言，根据黎曼度量的空间各向异性，我们论证必须采用子黎曼近似。我们通过提供一种无论各向异性程度如何均适用的新近似核来解决此问题。我们提出了具有更优误差估计的新定理，并证明所有近似核均具有与精确核相同的反射对称性。我们在两个数据集上测试了PDE-G-CNN框架内多种近似核的有效性，观察到新近似核带来的改进。我们报告指出，PDE-G-CNNs在两个数据集上再次实现了网络复杂度的大幅降低，同时性能与G-CNNs和CNNs相当或更优。此外，由于形态学核与神经几何中的关联场相关，PDE-G-CNNs相比G-CNNs具有更好的几何可解释性优势。