Incidence Networks for Geometric Deep Learning
This work provides a foundational framework for geometric deep learning, enabling more efficient and structured processing of complex data types.
The paper formalizes incidence tensors for representing structured data like graphs and simplicial complexes, and presents a family of equivariant networks that operate on them, with an efficient implementation based on decomposition into invariant subsets.
Sparse incidence tensors can represent a variety of structured data. For example, we may represent attributed graphs using their node-node, node-edge, or edge-edge incidence matrices. In higher dimensions, incidence tensors can represent simplicial complexes and polytopes. In this paper, we formalize incidence tensors, analyze their structure, and present the family of equivariant networks that operate on them. We show that any incidence tensor decomposes into invariant subsets. This decomposition, in turn, leads to a decomposition of the corresponding equivariant linear maps, for which we prove an efficient pooling-and-broadcasting implementation.