Signal processing on simplicial complexes with vertex signals
In classical graph signal processing (GSP), the underlying topological structures are restricted in terms of dimensionality. A graph or a 1-complex is a combinatorial object that models binary relations, which do not directly capture complex high arity relations. One possible high dimensional genera...
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| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/164992 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | In classical graph signal processing (GSP), the underlying topological structures are restricted in terms of dimensionality. A graph or a 1-complex is a combinatorial object that models binary relations, which do not directly capture complex high arity relations. One possible high dimensional generalization of graphs is a simplicial complex. In this paper, we develop a signal processing framework on simplicial complexes with vertex signals, which recovers the traditional GSP theory. We introduce the concept of a generalized Laplacian, which allows us to embed a simplicial complex into a traditional graph and hence perform signal processing similar to traditional GSP. We show that the generalized Laplacian satisfies several desirable properties, same as the graph Laplacian. We propose a method to learn 2-complex structures and demonstrate how to perform signal processing by applying the generalized Laplacian on both synthetic and real data. We observe performance gains in our experiments when 2-complexes are used to model the data, compared to the traditional GSP approach of restricting to 1-complexes. |
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