Spectral Convergence of Complexon Shift Operators
This work addresses transferability issues for researchers in topological signal processing, offering a generalization of graphon frameworks, but it is incremental as it builds on existing complexon and graphon concepts.
The paper tackles the problem of transferability in Topological Signal Processing by constructing a complexon shift operator and proving that eigenvalues, eigenspaces, and Fourier transforms converge when simplicial complex signals approach a complexon limit, with verification through two numerical experiments.
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.