HIGH-HOPeS: Higher-Order Hodge Laplacians for Processing of multi-way Signals
ERC Starting Grant
(ERC StG – Grant agreement ID: 101039827)
This project has received funding by the European Union (ERC, HIGH-HOPeS, 101039827). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
Overview
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.
Presentations and Communication
Research related to the present project was presented at the following places / events.
- Autumn school on hypergraphs, Bernoulli Society Committee on Statistical Network Science, Online, October 2023
- ELLIIT Focus Period Network Dynamics and Control, Linköping, Sweden, September 2023
- International Congress on Industrial and Applied Mathematics (ICIAM 2023) – Minisymposium on Higher order networks for complex systems, Tokyo, Japan, August 2023
- Seminar at Chair of Data Analytics and Machine Learning, TUM, Munich, Germany, August 2023
- School lectures: Signal Processing on Networks, NetSci 2023, Vienna, Austria, July 2023
- Applications of Hodge Theory on Networks Workshop, Banff, Canada, February 2023
Complete Publication list
A list of all publications associated with this project is provided below:
- Hoppe, J. & Schaub, M. T., Representing Edge Flows on Graphs via Sparse Cell Complexes, Learning on Graphs 2023, 2023 (Best Paper Award)
- Grande, V. P. & Schaub, M. T., Non-isotropic Persistent Homology: Leveraging the Metric Dependency of PH, Learning on Graphs 2023, 2023
Contact
Computational Network Science Group
Department of Computer Science
RWTH Aachen University
Ahornstrasse 50
52074 Aachen
Germany