Before revision:  Our key insight is that a simple learned diffusion layer can spatially share data in a principled manner, replacing operations like convolution and pooling which are complicated and expensive on surfaces.
After revision:  Our key insight is that a simple diffusion layer is highly effective for spatial communication. The resulting networks automatically generalize across different samplings and resolutions of a surface -- a basic property which is crucial for practical applications. Our networks can be discretized on various geometric representations such as triangle meshes or point clouds, and can even be trained on one representation then applied to another. We optimize the spatial support of diffusion as a continuous network parameter ranging from purely local to totally global, removing the burden of manually choosing neighborhood sizes.