Multi-directional Geodesic Neural Networks via Equivariant Convolution
TimeThursday, 6 December 20182:36pm - 2:57pm
DescriptionWe propose a novel approach for performing convolution of signals defined on curved surfaces and show its utility in a variety of geometric deep learning applications. Key to our construction is the notion of angular functions defined on the surface, which extend the classic real-valued signals, and which can be naturally convolved 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 allows, in particular, to propagate directional information, after convolution, across layers and thus different regions on the shape, leading to the notion of multi-directional geodesic convolution, or directional convolution for short. We first define directional convolution in the continuous setting, prove its key properties and then show how it can be implemented in practice, for shapes represented as triangle meshes. We evaluate directional convolution in a wide variety of learning scenarios ranging from classification of signals on surfaces, to shape segmentation and shape matching, where we show a significant improvement over several baselines.