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000952482 001__ 952482 000952482 005__ 20230304053948.0 000952482 0247_ $$2CORDIS$$aG:(EU-Grant)101039827$$d101039827 000952482 0247_ $$2CORDIS$$aG:(EU-Call)ERC-2021-STG$$dERC-2021-STG 000952482 0247_ $$2originalID$$acorda_____he::101039827 000952482 035__ $$aG:(EU-Grant)101039827 000952482 150__ $$aHigher-Order Hodge Laplacians for Processing of multi-way Signals$$y2023-01-01 - 2027-12-31 000952482 372__ $$aERC-2021-STG$$s2023-01-01$$t2027-12-31 000952482 450__ $$aHIGH-HOPeS$$wd$$y2023-01-01 - 2027-12-31 000952482 5101_ $$0I:(DE-588b)5098525-5$$2CORDIS$$aEuropean Union 000952482 680__ $$aNetwork 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. 000952482 909CO $$ooai:juser.fz-juelich.de:1004911$$pauthority$$pauthority:GRANT 000952482 980__ $$aG 000952482 980__ $$aCORDIS 000952482 980__ $$aAUTHORITY