Multi-directional Geodesic Neural Networks via Equivariant Convolution

Understand

We propose a novel approach for performing convolution of signals on curved surfaces and show its utility in a variety of geometric deep learning applications.

• Key to our construction is the notion of directional functions defined on the surface, which extend the classic real-valued signals and which can be naturally convolved with with real-valued template functions.
• As a result, rather than trying to fix a canonical orientation or only keeping the maximal response across all alignments of a 2D template at every point of the surface, as done in previous works, we show how information across all rotations can be kept across different layers of the neural network.
• Our construction, which we call multi-directional geodesic convolution, or directional convolution for short, allows, in particular, to propagate and relate directional information across layers and thus different regions on the shape.