Signal Processing on Simplicial Complexes With Vertex Signals - Archive ouverte HAL
Journal Articles IEEE Access Year : 2022

Signal Processing on Simplicial Complexes With Vertex Signals

Abstract

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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hal-04412269 , version 1 (23-01-2024)

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Feng Ji, Giacomo Kahn, Wee Peng Tay. Signal Processing on Simplicial Complexes With Vertex Signals. IEEE Access, 2022, 10, pp.41889 - 41901. ⟨10.1109/ACCESS.2022.3167055⟩. ⟨hal-04412269⟩
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