Signal Processing on Cell Complexes

Abstract

The processing of signals supported on non-Euclidean domains has attracted large interest recently. Thus far, such non-Euclidean domains have been abstracted primarily as graphs with signals supported on the nodes, though the processing of signals on more general structures such as simplicial complexes has also been considered. In this paper, we give an introduction to signal processing on (abstract) regular cell complexes, which provide a unifying framework encompassing graphs, simplicial complexes, cubical complexes and various meshes as special cases. We discuss how appropriate Hodge Laplacians for these cell complexes can be derived. These Hodge Laplacians enable the construction of convolutional filters, which can be employed in linear filtering and non-linear filtering via neural networks defined on cell complexes.

arXiv comment5 pages, 3 figures

References in corpus (15)

• Semi-Supervised Classification with Graph Convolutional Networks
• The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains
• Networks beyond pairwise interactions: structure and dynamics
• Topological Signal Processing over Simplicial Complexes
• Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian
• Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond
• Toward a Spectral Theory of Cellular Sheaves
• Graph-based Semi-Supervised & Active Learning for Edge Flows
• Simplicial Neural Networks
• Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space
• Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks
• Cell Complex Neural Networks
• Finite Impulse Response Filters for Simplicial Complexes
• Simplicial 2-Complex Convolutional Neural Nets
• Simplicial Complex Representation Learning

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• Outlier Detection for Trajectories via Flow-embeddings
• Signal Processing on Product Spaces
• Topological Neural Networks over the Air
• Faster Inference of Cell Complexes from Flows via Matrix Factorization