Spectral Convergence of Complexon Shift Operators

TL;DR

The paper addresses transferability of topological signal processing on large or evolving simplicial complexes. It defines complexons as the limit objects of simplicial complex sequences and constructs a marginal complexon and a complexon shift operator (CSO) that operate across all dimensions. It proves that when a simplicial complex signal sequence converges to a complexon signal, the CSO's eigenvalues, eigenvectors, and Fourier transform converge to those of the limit complexon signal, linking them to a new family of weighted adjacency matrices. Two numerical experiments validate the theoretical convergence results, illustrating potential learning-transferability in high-order structures and extending graphon signal processing to 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.