Dist2Cycle: A Simplicial Neural Network for Homology Localization
Abstract
Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multiway ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher dimensional topological features of data, features to which graphs, encoding only pairwise relationships, remain oblivious. While attempts have been made to extend Graph Neural Networks (GNNs) to a simplicial complex setting, the methods do not inherently exploit, or reason about, the underlying topological structure of the network. We propose a graph convolutional model for learning functions parametrized by the $k$homological features of simplicial complexes. By spectrally manipulating their combinatorial $k$dimensional Hodge Laplacians, the proposed model enables learning topological features of the underlying simplicial complexes, specifically, the distance of each $k$simplex from the nearest "optimal" $k$th homology generator, effectively providing an alternative to homology localization.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 DOI:
 10.48550/arXiv.2110.15182
 arXiv:
 arXiv:2110.15182
 Bibcode:
 2021arXiv211015182D
 Keywords:

 Computer Science  Machine Learning;
 Mathematics  Algebraic Topology;
 55N31;
 I.2
 EPrint:
 9 pages, 5 figures