Signal processing on simplicial complexes with vertex signals

• Main Authors: Ji, Feng, Kahn, Giacomo, Tay, Wee Peng
• Other Authors: School of Electrical and Electronic Engineering
• Format: Article
• Language: English
• Published: 2023
• Subjects: Engineering::Electrical and electronic engineering; Graph Signal Processing; Multidimensional Data
• Online Access: https://hdl.handle.net/10356/164992
• Institution: Nanyang Technological University

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.