  Graph Convolutional Networks (GCNs) have shown to be effective in handling
unordered data like point clouds and meshes. In this work we propose novel
approaches for graph convolution, pooling and unpooling, inspired from finite
differences and algebraic multigrid frameworks. We form a parameterized
convolution kernel based on discretized differential operators, leveraging the
graph mass, gradient and Laplacian. This way, the parameterization does not
depend on the graph structure, only on the meaning of the network convolutions
as differential operators. To allow hierarchical representations of the input,
we propose pooling and unpooling operations that are based on algebraic
multigrid methods, which are mainly used to solve partial differential
equations on unstructured grids. To motivate and explain our method, we compare
it to standard convolutional neural networks, and show their similarities and
relations in the case of a regular grid. Our proposed method is demonstrated in
various experiments like classification and part-segmentation, achieving on par
or better than state of the art results. We also analyze the computational cost
of our method compared to other GCNs.
