Spectral Convergence of Complexon Shift Operators

TL;DR

This paper extends spectral analysis in Topological Signal Processing by introducing complexon shift operators, demonstrating their eigenvalue convergence, and exploring transferability in large simplicial complexes.

Contribution

It introduces a higher-order complexon shift operator, analyzes its spectral properties, and proves convergence results that generalize graphon-based signal processing.

Findings

Eigenvalues and eigenspaces of CSOs converge to those of the complexon signal.

Numerical experiments verify the spectral convergence.

Results suggest transferability of learning on large simplicial complexes.

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.