# Spectral Convergence of Complexon Shift Operators

arXiv: https://arxiv.org/abs/2309.07169 · HTML: https://mlsanity.com/paper/2309.07169/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 32 references that are in the map.
- First posted on arXiv 12 Sep 2023.

## Builds on (most cited first, 20 of 20)

- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Convolutional Neural Network Architectures for Signals Supported on Graphs](https://mlsanity.com/paper/1805.00165/) (2018)
- [Graph signal processing for machine learning: A review and new perspectives](https://mlsanity.com/paper/2007.16061/) (2020)
- [A Review of Graph Neural Networks and Their Applications in Power Systems](https://mlsanity.com/paper/2101.10025/) (2021)
- [Transferability of Graph Neural Networks: an Extended Graphon Approach](https://mlsanity.com/paper/2109.10096/) (2021)
- [Simplicial Convolutional Neural Networks](https://mlsanity.com/paper/2110.02585/) (2021)
- [Graphon Signal Processing](https://mlsanity.com/paper/2003.05030/) (2020)
- [Introducing Hypergraph Signal Processing: Theoretical Foundation and Practical Applications](https://mlsanity.com/paper/1907.09203/) (2019)
- [Graphon Filters: Graph Signal Processing in the Limit](https://mlsanity.com/paper/2003.02099/) (2020)
- [Learnable Hypergraph Laplacian for Hypergraph Learning](https://mlsanity.com/paper/2106.05701/) (2021)
- [SGAT: Simplicial Graph Attention Network](https://mlsanity.com/paper/2207.11761/) (2022)
- [Invariance-Preserving Localized Activation Functions for Graph Neural Networks](https://mlsanity.com/paper/1903.12575/) (2019)
- [The Graphon Fourier Transform](https://mlsanity.com/paper/1910.10195/) (2019)
- [Stability of Neural Networks on Manifolds to Relative Perturbations](https://mlsanity.com/paper/2110.04702/) (2021)
- [Signal processing on simplicial complexes](https://mlsanity.com/paper/2004.02392/) (2020)
- [A Hilbert Space Theory of Generalized Graph Signal Processing](https://mlsanity.com/paper/1904.11655/) (2019)
- [Graph Signal Processing over a Probability Space of Shift Operators](https://mlsanity.com/paper/2108.09192/) (2021)
- [Topological Signal Processing over Weighted Simplicial Complexes](https://mlsanity.com/paper/2302.08561/) (2023)

## Abstract (arXiv)

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.