Topological Signal Processing: An Application-Oriented Tutorial

2026-05-18

AI Summary

This work addresses the limitations of traditional graph signal processing, which is confined to node-level signals and thus unable to capture higher-order interactions inherent in complex systems. By leveraging simplicial complexes and combinatorial Hodge Laplacians, the study extends signal processing to higher-dimensional topological structures such as edges and triangles. It introduces a method for constructing higher-order signals from lagged node observations and develops a corresponding theory of topological Fourier transforms and filtering. Applied to brain imaging data, the proposed framework successfully uncovers nontrivial higher-order interaction patterns among sets of brain regions that are invisible to conventional approaches, thereby establishing a theoretical and practical bridge for higher-order topological signal processing.

Abstract

Many modern datasets are large and carry complex structural relationships. Graph-based methods have traditionally been used to represent networked data, modeling individual elements as nodes and pairwise interactions as edges. Furthermore, Graph Signal Processing (GSP) has been developed to analyze signals on graph nodes, such as temperature measurements (node signals) across different regions of a country represented as a graph. Topological Signal Processing (TSP) is an emerging field that generalizes GSP, enabling the analysis of signals defined not only on nodes but also on edges, triangles, and higher-dimensional network elements, modeled as simplicial complexes and related topological structures. This makes TSP naturally well-suited for studying higher-order interactions in complex systems by extending classical signal processing concepts, such as filtering and Fourier transforms, to the topological level. Despite its versatility, TSP remains challenging for many practitioners. Therefore, we present an accessible overview of TSP foundations while drawing connections with application-oriented settings. We focus on processing techniques based on the combinatorial Hodge Laplacian, which generalizes the graph Laplacian to simplicial complexes. In particular, we review key TSP concepts, relate them to real-world examples, and discuss how higher-order structures and signals can be derived from datasets. For instance, we introduce an edge-level signal capturing lagged interactions between nodal signals, and demonstrate its use in a case study on TSP-based analysis of brain imaging data, revealing nontrivial interactions between sets of brain regions. Overall, we aim to promote a broader adoption of TSP by bridging methodological developments with applications, fostering its use among a wide community of theoretical and applied researchers.

Problem

• Topological Signal Processing
• Higher-order interactions
• Simplicial complexes
• Graph Signal Processing
• Hodge Laplacian

Innovation

• Topological Signal Processing
• Simplicial Complexes
• Hodge Laplacian
• Higher-order Interactions
• Graph Signal Processing