Dist2Cycle: A Simplicial Neural Network for Homology Localization

Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all once. This concept is central towards detection higher topological features data, to which graphs, encoding only pairwise relationships, remain oblivious. While attempts have been made extend Graph Neural Networks (GNNs) a simplicial complex setting, the methods do not inherently exploit, or reason about, underlying structure network. We propose graph convolutional model for learning functions parametrized by k-homological complexes. By spectrally manipulating their combinatorial k-dimensional Hodge Laplacians, proposed enables complexes, specifically, distance each k-simplex from nearest "optimal" k-th homology generator, effectively providing an alternative localization.