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.

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generalized spectral domain example workflowFigure 1: We present a method to simplify geometry while preserving targeted bandwidths in a customizable spectral domain.The input to our algorithm is a generalized spectral domain, a combination of spectral bands from multiple discrete Laplacian operators. Our method includes a simple łliftingž operator to enable the workflow shown on the right. The example shows diffusion distances over a tetrahedral mesh of an engine part with 20 small holes that are preserved. Removing 80% of the input simplices, and seeking to preserve topology (lower portion of the spectrum), leads to some visible differences with respect to the ground truth but the quantitative comparisons indicate low error (≈ 3.6𝑒 − 7, shown in the supplemental material).

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Reducing the size of triangle meshes and their higher-dimensional counterparts, called simplicial complexes, while preserving important geometric or topological properties is an important problem in computer graphics and geometry processing. Such salient properties are captured by local shape descriptors via linear differential operators -often variants of Laplacian matrices. The eigenfunctions of Laplacians yield a convenient and useful set of bases that define a spectral domain for geometry processing (akin to the famous Fourier spectrum which uses eigenfunctions of the derivative operator). Existing methods for spectrum-preserving coarsening focus on 0-dimensional Laplacian operators that are defined on vertices (0-dimensional simplices).We propose a generalized spectral coarsening method that considers multiple Laplacian operators of possibly different dimensionalities in tandem. Our simple algorithm greedily decides the order of contractions of simplices based on a quality function per simplex. The quality function quantifies the error due to removal of that simplex on a chosen band within the spectrum of the coarsened geometry. We demonstrate that our method is useful to achieve band-pass filtering on both meshes as well as general simplicial complexes. CCS CONCEPTS• Computing methodologies → Shape analysis.