Dist2Cycle: A Simplicial Neural Network for Homology Localization
This addresses the challenge of exploiting topological structure in neural networks for data analysis, offering a method to detect higher-dimensional features that graphs miss, though it appears incremental as an extension of GNNs to simplicial settings.
The paper tackles the problem of learning topological features in simplicial complexes, which generalize graphs to encode multi-way relations, by proposing a graph convolutional model that uses Hodge Laplacians to compute distances of simplices to homology generators, effectively enabling 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 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.