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In this work, we introduce a novel approach based on algebraic topology toenhance graph convolution and attention modules by incorporating localtopological properties of the data. To do so, we consider the framework ofsheaf neural networks, which has been previously leveraged to incorporateadditional structure into graph neural networks' features and construct moreexpressive, non-isotropic messages. Specifically, given an input simplicialcomplex (e.g. generated by the cliques of a graph or the neighbors in a pointcloud), we construct its local homology sheaf, which assigns to each node thevector space of its local homology. The intermediate features of our networkslive in these vector spaces and we leverage the associated sheaf Laplacian toconstruct more complex linear messages between them. Moreover, we extend thisapproach by considering the persistent version of local homology associatedwith a weighted simplicial complex (e.g., built from pairwise distances ofnodes embeddings). This i) solves the problem of the lack of a natural choiceof basis for the local homology vector spaces and ii) makes the sheaf itselfdifferentiable, which enables our models to directly optimize the topology oftheir intermediate features.

Algebraic Topological Networks via the Persistent Local Homology Sheaf