On Optimal Lifting to SE(2) in Equivariant Neural Networks

Equivariant neural networks, such as PDE-GCNNs, often require data as feature maps over the group. In this paper, we motivate criteria for the optimal lifting of feature maps over $\mathbb{R}^2$ to SE(2). We propose three optimality criteria: fast reconstruction property which ensures that no information is lost during the lifting, spatial locality and orientation locality. The locality conditions make sure that information is organized in a meaningful manner in the lifted space. We formulate corresponding losses which we then numerically minimize, and find that kernels emerge that closely resemble cake wavelets. The results indicate that the cake wavelets are near optimal under presented criteria, and as such provide an excellent starting point for any SE(2) equivariant architecture.