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HIGH-HOPeS

Higher-Order Hodge Laplacians for Processing of multi-way Signals

Grant period2023-01-01 - 2027-12-31
Funding bodyEuropean Union
Call numberERC-2021-STG
Grant number101039827
IdentifierG:(EU-Grant)101039827

Note: Network analysis has revolutionized our understanding of complex systems, and graph-based methods have emerged as powerful tools to process signals on non-Euclidean domains via graph signal processing and graph neural networks. The graph Laplacian and related matrices are pivotal to such analyses: i) the Laplacian serves as algebraic descriptor of the relationships between nodes; moreover, it is key for the analysis of network structure, for local operations such as averaging over connected nodes, and for network dynamics like diffusion and consensus; ii) Laplacian eigenvectors are natural basis-functions for data on graphs and endowed with meaningful variability notions for graph signals, akin to Fourier analysis in Euclidean domains. However, graphs are ill-equipped to encode multi-way and higher-order relations that are becoming increasingly important to comprehend complex datasets and systems in many applications, e.g., to understand group-dynamics in social systems, multi-gene interactions in genetic data, or multi-way drug interactions. The goal of this project is to develop methods that can utilize such higher-order relations, going from mathematical models to efficient algorithms and software. Specifically, we will focus on ideas from algebraic topology and discrete calculus, according to which the graph Laplacian can be seen as part of a hierarchy of Hodge-Laplacians that emerge from treating graphs as instances of more general cell complexes that systematically encode couplings between node-tuples of any size. Our ambition is to i) provide more informative ways to represent and analyze the structure of complex systems, paying special attention to computational efficiency; ii) translate the success of graph-based signal processing to data on general topological spaces defined by cell complexes; and iii) by generalizing from graphs to neural networks on complexes, gain deeper theoretical insights on the principles of graph neural networks as special case.
   

Recent Publications

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http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Preprint  ;  ;  ;  ;  ;
TopoEmbedX: A General Framework for Representation Learning on Topological Domains
() [10.48550/arXiv.2609.37884]  GO BibTeX | EndNote: XML, Text | RIS

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Preprint  ;
A repository for discovery and reuse of higher-order network datasets
() [10.48550/arXiv.2608.03491]  GO BibTeX | EndNote: XML, Text | RIS

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Hypergraphs and simplicial complexes in focus: a roadmap for future research in higher-order interactions
Journal of physics / Complexity 7(2), 022501 () [10.1088/2632-072X/ae3c4e]  GO OpenAccess  Download fulltext Files BibTeX | EndNote: XML, Text | RIS

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Contribution to a book/Contribution to a conference proceedings  ;
Point-Level Topological Representation Learning on Point Clouds
International Conference on Machine Learning, 13-19 July 2025, Vancouver Convention Center, Vancouver, Canada / Editors: Aarti Singh, Maryam Fazel, Daniel Hsu, Simon Lacoste-Julien, Felix Berkenkamp, Tegan Maharaj, Kiri Wagstaff, Jerry Zhu
42. International Conference on Machine Learning, ICML 2025, VancouverVancouver, Canada, 13 Jul 2025 - 19 Jul 20252025-07-132025-07-19 MLResearchPress, Proceedings of machine learning research 267, 20368-20398 ()  GO   Download fulltextHomepage of book Download fulltextFulltext BibTeX | EndNote: XML, Text | RIS

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Faster Inference of Cell Complexes from Flows via Matrix Factorization
5 Seiten () [10.48550/arXiv.2508.21372]  GO arXiv  Download fulltext Files  Download fulltextFulltext by arXiv.org BibTeX | EndNote: XML, Text | RIS

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Contribution to a book/Contribution to a conference proceedings  ;  ;
Faster Inference of Cell Complexes from Flows via Matrix Factorization
2025 33rd European Signal Processing Conference (EUSIPCO) : [Proceedings]
33. European Signal Processing Conference, EUSIPCO 2025, PalermoPalermo, Italy, 8 Sep 2025 - 12 Sep 20252025-09-082025-09-12 IEEE 2487-2491 () [10.23919/EUSIPCO63237.2025.11226659]  GO OpenAccess  Download fulltext Files BibTeX | EndNote: XML, Text | RIS

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Selecting fitted models under epistemic uncertainty using a stochastic process on quantile functions
Nature Communications 16, 9393 () [10.1038/s41467-025-64658-7]  GO OpenAccess  Download fulltext Files BibTeX | EndNote: XML, Text | RIS

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Don't be Afraid of Cell Complexes : An Introduction from an Applied Perspective
41 Seiten () [10.48550/arXiv.2506.09726]  GO arXiv  Download fulltext Files  Download fulltextFulltext by arXiv.org BibTeX | EndNote: XML, Text | RIS

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HLSAD: Hodge Laplacian-based Simplicial Anomaly Detection
11 Seiten () [10.48550/arXiv.2505.24534]  GO OpenAccess  Download fulltext Files  Download fulltextFulltext by arXiv.org BibTeX | EndNote: XML, Text | RIS

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Contribution to a book/Contribution to a conference proceedings  ;
A Bayesian Perspective on Uncertainty Quantification for Estimated Graph Signals
2025 IEEE International Conference on Acoustics, Speech, and Signal Processing : conference proceedings : Hyderabad, India / sponsors and organizer: IEEE, IEEE Signal Processing Society ; K V S Hari, V John Mathews, general chairs
2025 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2025, HyderabadHyderabad, India, 6 Apr 2025 - 11 Apr 20252025-04-062025-04-11 [Piscataway, NJ] : IEEE 5 Seiten () [10.1109/ICASSP49660.2025.10889783]  GO  Download fulltext Files BibTeX | EndNote: XML, Text | RIS

All known publications ...
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 Record created 2023-03-03, last modified 2023-03-04
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