Spectral Convergence of Simplicial Complex Signals: Transferability of Topological Signal Processing

核心概念

The core message of this work is to propose a novel framework called complexon shift operator (CSO) and derive its transferability properties, as a limit theory of simplicial complexes. This paves the way for complexon-based signal processing, making it a viable tool for analyzing signals on large and dynamic simplicial complex structures.

摘要

The authors introduce the concept of a complexon, which is the limit of a sequence of simplicial complexes, and define the complexon shift operator (CSO) as a marginal complexon. They investigate the relationship between the CSO and a family of adjacency matrices, referred to as raised adjacency matrices. The authors prove the transferability of simplicial complex signal processing by showing that if a sequence of simplicial complexes converges to a complexon, then the eigenvalues of their induced CSOs also converge. Furthermore, they derive the convergence of the complexon signal Fourier transform under certain conditions. The results are verified through numerical experiments.
The key highlights and insights are:

Proposal of the CSO concept for complexons, analogous to the graphon shift operator for graphons.
Investigation of the raised adjacency matrix for simplicial complexes and its relation to the CSO of the induced complexon.
Derivation of the transferability properties of the CSO, including the convergence of eigenvalues.
Derivation of the convergence of complexon signal Fourier transform under specific conditions.
Verification of the theoretical results through numerical experiments on synthetic data.

統計資料

The authors do not provide any specific numerical data or metrics in the content. The focus is on the theoretical development and analysis of the complexon signal processing framework.

引述

"We propose the concept of a CSO for complexons, analogous to the graphon shift operator for graphons."
"We derive the transferability properties of the CSO and use a numerical experiment to verify it."
"We derive the convergence of complexon signal Fourier transform and use a mathematical model to illustrate it."

從以下內容提煉的關鍵洞見

Spectral Convergence of Simplicial Complex Signals

by Purui Zhang,... 於 arxiv.org 04-09-2024

https://arxiv.org/pdf/2309.07169.pdf

深入探究

How can the proposed complexon signal processing framework be extended to handle more complex real-world scenarios, such as dynamic or heterogeneous simplicial complex structures

The proposed complexon signal processing framework can be extended to handle more complex real-world scenarios by incorporating dynamic or heterogeneous simplicial complex structures. One approach could involve developing adaptive algorithms that can adjust the processing parameters based on the evolving topology of the complex. For dynamic structures, the framework could include mechanisms to detect changes in the complex and update the signal processing accordingly. Additionally, for heterogeneous structures, the framework could incorporate methods to handle varying properties or dimensions within the complex. By integrating these capabilities, the complexon-based signal processing approach can effectively adapt to a wide range of real-world scenarios, making it more versatile and robust.

What are the potential applications and practical implications of the complexon-based signal processing approach beyond the theoretical developments presented in this work

The potential applications and practical implications of the complexon-based signal processing approach extend beyond the theoretical developments presented in this work. One key application is in the analysis of large-scale network data, where complex structures can be effectively modeled and analyzed using the complexon framework. This can have significant implications in various fields such as social network analysis, biological network modeling, and communication networks. The framework can also be applied in image processing, where complex spatial relationships can be captured and processed using simplicial complexes. Furthermore, in the field of sensor networks, the complexon-based approach can enable efficient processing of multi-dimensional sensor data, leading to improved data analysis and decision-making.

Can the convergence properties of the complexon Fourier transform be further generalized or relaxed to accommodate a wider range of complexon signals and applications

The convergence properties of the complexon Fourier transform can be further generalized or relaxed to accommodate a wider range of complexon signals and applications by considering different signal characteristics and properties. One way to generalize the convergence properties is to explore the convergence behavior under varying signal distributions or noise levels. By studying the convergence under different conditions, the framework can be adapted to handle a broader spectrum of signal variations. Additionally, relaxing the convergence criteria to allow for approximate convergence or partial convergence can make the framework more flexible and applicable to a wider range of practical scenarios. This flexibility can enhance the utility of the complexon-based signal processing approach in diverse real-world applications.