Spectral Convergence of Complexon Shift Operators

• URL: http://arxiv.org/abs/2309.07169v4
• Date: Sun, 5 May 2024 10:52:26 GMT
• Title: Spectral Convergence of Complexon Shift Operators
• Authors: Purui Zhang, Xingchao Jian, Feng Ji, Wee Peng Tay, Bihan Wen,
• Abstract summary: We study the transferability of Topological Signal Processing via a generalized higher-order version of graphon, known as complexon. Inspired by the graphon shift operator and message-passing neural network, we construct a marginal complexon and complexon shift operator. We prove that when a simplicial complex signal sequence converges to a complexon signal, the eigenvalues, eigenspaces, and Fourier transform of the corresponding CSOs converge to that of the limit complexon signal.
• Score: 38.89310649097387
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Topological Signal Processing (TSP) utilizes simplicial complexes to model structures with higher order than vertices and edges. In this paper, we study the transferability of TSP via a generalized higher-order version of graphon, known as complexon. We recall the notion of a complexon as the limit of a simplicial complex sequence [1]. Inspired by the graphon shift operator and message-passing neural network, we construct a marginal complexon and complexon shift operator (CSO) according to components of all possible dimensions from the complexon. We investigate the CSO's eigenvalues and eigenvectors and relate them to a new family of weighted adjacency matrices. We prove that when a simplicial complex signal sequence converges to a complexon signal, the eigenvalues, eigenspaces, and Fourier transform of the corresponding CSOs converge to that of the limit complexon signal. This conclusion is further verified by two numerical experiments. These results hint at learning transferability on large simplicial complexes or simplicial complex sequences, which generalize the graphon signal processing framework.

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