Dist2Cycle: A Simplicial Neural Network for Homology Localization

TL;DR

This paper introduces Dist2Cycle, a novel spectral simplicial neural network that learns topological features of data by analyzing the homology of simplicial complexes, enabling effective homology localization.

Contribution

It proposes a spectral graph convolutional model that leverages Hodge Laplacians to learn homological features, advancing topological data analysis with neural networks.

Findings

Successfully localizes homology generators in complex data.

Outperforms existing methods in topological feature detection.

Provides a new neural approach for topological analysis.

Abstract

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.