（次）黎曼 PDE-G-CNNs 的分析
Analysis of (sub-)Riemannian PDE-G-CNNs
AI总结：
本文分析了PDE-G-CNNs中近似形态学核的精度问题，提出了一种适用于任意空间各向异性的新次黎曼近似核，并证明其在降低网络复杂度的同时提升了性能与几何可解释性。
摘要：
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训练本应硬编码到网络中的对称性上。最近引入的基于 PDE 的 G-CNNs（PDE-G-CNNs）框架推广了 G-CNNs。PDE-G-CNNs 的核心优势是它们同时 1) 降低网络复杂度，2) 提高分类性能，以及 3) 提供几何可解释性。它们的实现主要由带有核的线性和形态学卷积组成。在本文中，我们表明先前提出的近似形态学核并不总是能准确逼近精确核。更具体地说，根据黎曼度量的空间各向异性，我们认为必须求助于次黎曼近似。我们通过提供一种无论各向异性如何都能起作用的新近似核来解决这个问题。我们提供了具有更好近似核误差估计的新定理，并证明它们都带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 框架内多个近似核的有效性，并观察到使用新近似核的改进。我们报告称，PDE-G-CNNs 再次允许大幅降低网络复杂度，同时在两个数据集上具有与 G-CNNs 和 CNNs 相当或更好的性能。此外，PDE-G-CNNs 相比 G-CNNs 具有更好几何可解释性的优势，因为形态学核与神经几何学中的关联场相关。
英文摘要：
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.
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