Abstract:
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.
