Higher-Order Hodge Laplacians for Processing of multi-way Signals
Grant period
2023-01-01 - 2027-12-31
Funding body
European Union
Call number
ERC-2021-STG
Grant number
101039827
Identifier
G:(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.
http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.pngContribution to a book/Contribution to a conference proceedingsRompelberg, L. J. J. (Corresponding author)RWTH* ; Schaub, M. T. (Corresponding author)RWTH* A Bayesian Perspective on Uncertainty Quantification for Estimated Graph Signals 20252025 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] : IEEE5 Seiten(2025)[10.1109/ICASSP49660.2025.10889783]2025 FilesBibTeX |
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http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.pngContribution to a book/Contribution to a conference proceedingsEpping, B. (Corresponding author)RWTH* ; René, A. (Corresponding author)RWTH* ; Helias, M. (Corresponding author)RWTH* ; Schaub, M. T. (Corresponding author)RWTH* Graph Neural Networks Do Not Always Oversmooth 2024Advances in Neural Information Processing Systems 37 (NeurIPS 2024) / edited by: A. Globerson and L. Mackey and D. Belgrave and A. Fan and U. Paquet and J. Tomczak and C. Zhang 38. Conference on Neural Information Processing Systems, NeurIPS 2024, VancouverVancouver, Canada, 10 Dec 2024 - 15 Dec 20242024-12-102024-12-15[Erscheinungsort nicht ermittelbar] : [Verlag nicht ermittelbar], Advances in neural information processing systems37,[1]-25(2024)2024FulltextHomepage of bookBibTeX |
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http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.pngContribution to a book/Contribution to a conference proceedingsGrande, V. P. (Corresponding author)RWTH* ; Hoppe, J. (Corresponding author)RWTH* ; Frantzen, F. (Corresponding author)RWTH* ; Schaub, M. T. (Corresponding author)RWTH* Topological Trajectory Classification and Landmark Inference on Simplicial Complexes 2024Conference record of the Fifty-Eighth Asilomar Conference on Signals, Systems & Computers : October 27 - 30, 2024, Pacific Grove, California / edited by: Michael B. Matthews, ICR ; technical co-sponsor: the IEEE Signal Processing Society 58. Asilomar Conference on Signals, Systems, and Computers, Pacific GrovePacific Grove, USA, 27 Oct 2024 - 30 Oct 20242024-10-272024-10-30[Piscataway, NJ] : IEEE44-48(2024)[10.1109/IEEECONF60004.2024.10942887]2024BibTeX |
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