Geometric Adaptations of PDE-G-CNNs

Group equivariant convolutional neural networks (G-CNNs) have been successfully applied in geometric deep learning. The recently introduced framework of PDE-based G-CNNs (PDE-G-CNNs) generalizes G-CNNs while simultaneously reducing network complexity and increasing performance. In PDE-G-CNNs the usual building blocks of neural networks are replaced with solvers for evolution PDEs, these PDEs being convection, diffusion, dilation, and erosion. We investigate three geometric adaptations of PDE-G-CNNs: We generalize the theory in [2] to a family of Lie groups in between roto-translation group SE(2) and Heisenberg group H(3). This geometric adaptation enables transferring training orientation score processing on SE(2) to training processing of velocity scores, shearlet transforms, or frequency scores on H(3).We theoretically prove that the trainable lifting layer in a PDE-G-CNN is interchangeable with a single fixed untrained lifting coupled with multiple trainable convections. We experimentally validate this theoretical insight and report identical performance. This fixing of the lifting layer makes PDE-G-CNNs more interpretable as they now solely train association fields from neurogeometry.We include curvature adaptation in PDE-G-CNNs. This curvature adaptation is beneficial within the convection part of PDE-G-CNNs as we show experimentally.