Background of GNNs and Simplicial Complexes
A graph is a type of data structure that consists of two components, vertices, and edges. Graph Neural Network (GNN) is a neural network type that operates on a Graph base. An example of Graph Neural Network (GNN) is node classification. (1)
A simplicial complex is a set of points, line segments, and triangles. Triangulation in this space forms the basis for constructing all other things like n-dimensional counterparts or new ones with different dimensions.(2,3)
Understanding the Problem
One way to understand simplicial complexes is by viewing them as high-dimensional generalizations of graphs that explicitly encode multiway ordered relations between vertices at different resolutions. This feature is vital to detecting higher-dimensional topological relationships in data, but graphs only capture pairwise connections. Though there have been many attempts to extend GNNs to simplicial complexes, the methods do not naturally exploit or reason about this network’s underlying topological structure.
Primary Focus of this paper: This paper flips the conventional view of approximation, which is to say it guides learners on simplicial complexes.
Researchers from The University of Edinburgh and Oxford propose a GNN model (Dist2Cycle) for localizing homological information in the form of distance functions between each point on complex to its nearest homology generating feature.
Since homology-related information is contained in its null space, the research team focused on the subspace spanned by eigenvectors corresponding to the lowest values (eigenvalues) instead of using the most significant ones (eigenvectors) of the relevant Laplacian operator. The research group implemented this method by calculating the most-significant subspace of an inverted version of their operator, which they believe is more optimal than traditional ones.
The proposed model via GNN showcases simplicial complexes as computational graphs for inference, also called the Hodge Laplacian graphs. The research paper explains the structure and benefits of this new homology-aware graph convolution framework operating on Hodge Laplacian graphs, taking advantage of its spectral properties of the Hodge Laplacian.