  We introduce a new general-purpose approach to deep learning on 3D surfaces,
based on the insight that a simple diffusion layer is highly effective for
spatial communication. The resulting networks are automatically robust to
changes in resolution and sampling 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. The only other ingredients in the method are a multi-layer
perceptron applied independently at each point, and spatial gradient features
to support directional filters. The resulting networks are simple, robust, and
efficient. Here, we focus primarily on triangle mesh surfaces, and demonstrate
state-of-the-art results for a variety of tasks including surface
classification, segmentation, and non-rigid correspondence.
