Topological signal processing and learning: Recent advances and future challenges

Developing methods to process irregularly structured data is crucial in applications like gene-regulatory, brain, power, and socioeconomic networks. Graphs have been the go-to algebraic tool for modeling the structure via nodes and edges capturing their interactions, leading to the establishment of...

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Bibliographic Details
Published in:Signal processing Vol. 233; p. 109930
Main Authors: Isufi, Elvin, Leus, Geert, Beferull-Lozano, Baltasar, Barbarossa, Sergio, Di Lorenzo, Paolo
Format: Journal Article
Language:English
Published: Elsevier B.V 01.08.2025
Subjects:
Graph machine learning
Graph signal processing
Hodge theory
Network science
Topological data analysis
Topological deep learning
Topological signal processing
Graph machine learning
Topological signal processing
Hodge theory
Topological data analysis
Graph signal processing
Network science
Topological deep learning
ISSN:0165-1684
Online Access:Get full text
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Summary:Developing methods to process irregularly structured data is crucial in applications like gene-regulatory, brain, power, and socioeconomic networks. Graphs have been the go-to algebraic tool for modeling the structure via nodes and edges capturing their interactions, leading to the establishment of the fields of graph signal processing (GSP) and graph machine learning (GML). Key graph-aware methods include Fourier transform, filtering, sampling, as well as topology identification and spatiotemporal processing. Although versatile, graphs can model only pairwise dependencies in the data. To this end, topological structures such as simplicial and cell complexes have emerged as algebraic representations for more intricate structure modeling in data-driven systems, fueling the rapid development of novel topological-based processing and learning methods. This paper first presents the core principles of topological signal processing through the Hodge theory, a framework instrumental in propelling the field forward thanks to principled connections with GSP-GML. It then outlines advances in topological signal representation, filtering, and sampling, as well as inferring topological structures from data, processing spatiotemporal topological signals, and connections with topological machine learning. The impact of topological signal processing and learning is finally highlighted in applications dealing with flow data over networks, geometric processing, statistical ranking, biology, and semantic communication.
ISSN:0165-1684
DOI:10.1016/j.sigpro.2025.109930