Analysis of (sub-)Riemannian PDE-G-CNNs
This work addresses a technical bottleneck in geometric deep learning for researchers and practitioners using PDE-G-CNNs, offering an incremental improvement in kernel approximation accuracy.
The paper tackled the problem of inaccurate approximations in morphological kernels within PDE-based group equivariant convolutional neural networks (PDE-G-CNNs), particularly under anisotropic Riemannian metrics, by introducing a new approximative kernel that works regardless of anisotropy and providing better error estimates. The result showed that PDE-G-CNNs with the new kernels achieved comparable or better performance than G-CNNs and CNNs on two datasets while reducing network complexity.
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.