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.

翻译：拓扑信号处理利用单纯复形来建模高于顶点和边的结构。本文通过图极限的一种广义高阶版本——复形极限，研究拓扑信号处理的可迁移性。我们回顾了复形极限作为单纯复形序列极限的概念[1]。受图移位算子和消息传递神经网络的启发，我们根据复形极限中所有可能维度的分量构建了边际复形极限和复形极限移位算子。我们研究了CSO的特征值和特征向量，并将其与一类新的加权邻接矩阵相关联。我们证明：当单纯复形信号序列收敛于复形极限信号时，对应CSO的特征值、特征空间和傅里叶变换也收敛于极限复形信号的特征值、特征空间和傅里叶变换。这一结论通过两个数值实验得到进一步验证。这些结果揭示了在大规模单纯复形或单纯复形序列上学习可迁移性的可能，从而推广了图信号处理框架。