Frequency Convergence of Complexon Shift Operators (Extended Version)
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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. Inspired by the integral operator form of graphon shift operators, 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 sequence converges to a complexon, the eigenvalues of the corresponding CSOs converge to that of the limit complexon. This conclusion is further verified by a numerical experiment. 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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