# Topological Signal Processing: An Application-Oriented Tutorial

arXiv: https://arxiv.org/abs/2605.22853 · HTML: https://mlsanity.com/paper/2605.22853/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 48 of its 96 references that are in the map.
- First posted on arXiv 18 May 2026.

## Builds on (most cited first, 30 of 48)

- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Topological Deep Learning: Going Beyond Graph Data](https://mlsanity.com/paper/2206.00606/) (2022)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [What are higher-order networks?](https://mlsanity.com/paper/2104.11329/) (2021)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Simplicial Convolutional Neural Networks](https://mlsanity.com/paper/2110.02585/) (2021)
- [Graph Signal Processing: History, Development, Impact, and Outlook](https://mlsanity.com/paper/2303.12211/) (2023)
- [Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space](https://mlsanity.com/paper/1808.02111/) (2018)
- [Signal Processing on Cell Complexes](https://mlsanity.com/paper/2110.05614/) (2021)
- [Convolutional Learning on Simplicial Complexes](https://mlsanity.com/paper/2301.11163/) (2023)
- [Generalized Simplicial Attention Neural Networks](https://mlsanity.com/paper/2309.02138/) (2023)
- [Topological Signal Processing over Cell Complexes](https://mlsanity.com/paper/2112.06709/) (2021)
- [Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2303.11323/) (2023)
- [Introducing Hypergraph Signal Processing: Theoretical Foundation and Practical Applications](https://mlsanity.com/paper/1907.09203/) (2019)
- [Simplicial Convolutional Filters](https://mlsanity.com/paper/2201.11720/) (2022)
- [Simplicial Representation Learning with Neural k-Forms](https://mlsanity.com/paper/2312.08515/) (2023)
- [Unveiling the higher-order organization of multivariate time series](https://mlsanity.com/paper/2203.10702/) (2022)
- [Hodge-Compositional Edge Gaussian Processes](https://mlsanity.com/paper/2310.19450/) (2023)
- [Signal processing on simplicial complexes](https://mlsanity.com/paper/2106.07471/) (2021)
- [Topological Signal Processing and Learning: Recent Advances and Future Challenges](https://mlsanity.com/paper/2412.01576/) (2024)
- [The topological Dirac equation of networks and simplicial complexes](https://mlsanity.com/paper/2106.02929/) (2021)
- [Hodgelets: Localized Spectral Representations of Flows on Simplicial Complexes](https://mlsanity.com/paper/2109.08728/) (2021)
- [Decoupling of brain function from structure reveals regional behavioral specialization in humans](https://mlsanity.com/paper/1905.07813/) (2019)
- [Topological Signal Processing over Generalized Cell Complexes](https://mlsanity.com/paper/2201.08993/) (2022)
- [Dirac signal processing of higher-order topological signals](https://mlsanity.com/paper/2301.10137/) (2023)

## Abstract (arXiv)

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.