Analysis of (sub-)Riemannian PDE-G-CNNs

Abstract

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.

PDE-CNNs: Axiomatic Derivations and Applications

“…These results align with the conclusions in [1,[4][5][6] where PDE-G-CNNs and G-CNNs on the roto-translation group G = SE(2) are compared.…”

“…x dx also satisfies Definition 25. However, and this is also mentioned in [25], this transform is limited in its applicability because it is only finitely-valued for functions with super-exponential decay 6 . Given these limitation of this transform, we instead use the normal Fourier transform.…”

“…Using the definition of the translation operator T v (6) and the integral operator axiom we expand this into:…”

“…It is shown in [1,[4][5][6] that PDE-G-CNNs -in addition to being inherently equivariant -require fewer parameters, have increased performance, are more data efficient, and more geometrically interpretable, when compared to G-CNNs and classical CNNs. Regarding the geometric interpretability, there is a relation between PDE-G-CNNs and Fig.…”

“…The PDE-G-CNN architecture is very general in the sense that the signals can be defined on any Lie group G. However, the existing PDE-G-CNN literature [1,[4][5][6] mainly concerns itself with G = SE (2), the group of two-dimensional rotations and translations. We will not consider the general setting, or G = SE (2) for that matter, and restrict ourselves to G = R 2 , i.e.…”