Algebraic Topological Networks via the Persistent Local Homology Sheaf
Gabriele Cesa Arash Behboodi
In this work, we introduce a novel approach based on algebraic topology to
enhance graph convolution and attention modules by incorporating local
topological properties of the data. To do so, we consider the framework of
sheaf neural networks, which has been previously leveraged to incorporate
additional structure into graph neural networks' features and construct more
expressive, non-isotropic messages. Specifically, given an input simplicial
complex (e.g. generated by the cliques of a graph or the neighbors in a point
cloud), we construct its local homology sheaf, which assigns to each node the
vector space of its local homology. The intermediate features of our networks
live in these vector spaces and we leverage the associated sheaf Laplacian to
construct more complex linear messages between them. Moreover, we extend this
approach by considering the persistent version of local homology associated
with a weighted simplicial complex (e.g., built from pairwise distances of
nodes embeddings). This i) solves the problem of the lack of a natural choice
of basis for the local homology vector spaces and ii) makes the sheaf itself
differentiable, which enables our models to directly optimize the topology of
their intermediate features.