HIGH-HOPeS

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

Funding

ERC Starting Grant , 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.
ERC Funded

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:

  1. 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;
  2. 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

  1. provide more informative ways to represent and analyze the structure of complex systems, paying special attention to computational efficiency;
  2. translate the success of graph-based signal processing to data on general topological spaces defined by cell complexes; and
  3. by generalizing from graphs to neural networks on complexes, gain deeper theoretical insights on the principles of graph neural networks as special case.

Presentation and Communication

Research related to the present project was presented by M. Schaub at the following places / events:

Publications

In association with HIGH-HOPeS, we have published the following works:

Preprint

  • Hoppe, J., Grande, V. P., & Schaub, M. T. (2025). Don’t be Afraid of Cell Complexes! An Introduction from an Applied Perspective (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2506.09726
  • Patel, D., Savostianov, A., & Schaub, M. T. (2025). Convergence of gradient based training for linear Graph Neural Networks (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2501.14440
  • Savostianov, A., Schaub, M. T., Guglielmi, N., & Tudisco, F. (2025). Efficient Sparsification of Simplicial Complexes via Local Densities of States (Version 2). arXiv. https://doi.org/10.48550/ARXIV.2502.07558
  • Hoppe, J., & Schaub, M. T. (2024). Random Abstract Cell Complexes (Version 2). arXiv. https://doi.org/10.48550/ARXIV.2406.01999
  • Telyatnikov, L., Bernardez, G., Montagna, M., Hajij, M., Carrasco, M., Vasylenko, P., Papillon, M., Zamzmi, G., Schaub, M. T., Verhellen, J., Snopov, P., Miquel-Oliver, B., Gil-Sorribes, M., Molina, A., Guallar, V., Long, T., Suk, J., Rygiel, P., Nikitin, A., … Papamarkou, T. (2024). TopoBench: A Framework for Benchmarking Topological Deep Learning (Version 3). arXiv. https://doi.org/10.48550/ARXIV.2406.06642
  • Hajij, M., Zamzmi, G., Papamarkou, T., Miolane, N., Guzmán-Sáenz, A., Ramamurthy, K. N., Birdal, T., Dey, T. K., Mukherjee, S., Samaga, S. N., Livesay, N., Walters, R., Rosen, P., & Schaub, M. T. (2022). Topological Deep Learning: Going Beyond Graph Data (Version 3). arXiv. https://doi.org/10.48550/ARXIV.2206.00606

2025

  • Spreuer, T., Hoppe, J., & Schaub, M. T. (2025). Faster Inference of Cell Complexes from Flows via Matrix Factorization. 2025 33rd European Signal Processing Conference (EUSIPCO), 2487–2491. https://doi.org/10.23919/eusipco63237.2025.11226659
    [ PDF ] [ Slides ] [ Code ]
  • Frantzen, F., & Schaub, M. T. (2025). HLSAD: Hodge Laplacian-based Simplicial Anomaly Detection. Proceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining V.2, 626–636. https://doi.org/10.1145/3711896.3736998
    [ Code ]
  • Grande, V. P., & Schaub, M. T. (2025). Point-Level Topological Representation Learning on Point Clouds. In A. Singh, M. Fazel, D. Hsu, S. Lacoste-Julien, F. Berkenkamp, T. Maharaj, K. Wagstaff, & J. Zhu (Eds.), Proceedings of the 42th International Conference on Machine Learning (Vol. 267, pp. 20368–20398). PMLR.
    [ PDF ] [ Code ]
  • Rompelberg, L., & Schaub, M. T. (2025). A Bayesian Perspective on Uncertainty Quantification for Estimated Graph Signals. ICASSP 2025 - 2025 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 1–5. https://doi.org/10.1109/icassp49660.2025.10889783
    [ PDF ] [ arXiv ]

2024

  • Papamarkou, T., Birdal, T., Bronstein, M. M., Carlsson, G. E., Curry, J., Gao, Y., Hajij, M., Kwitt, R., Lio, P., Di Lorenzo, P., Maroulas, V., Miolane, N., Nasrin, F., Natesan Ramamurthy, K., Rieck, B., Scardapane, S., Schaub, M. T., Veličković, P., Wang, B., … Zamzmi, G. (2024). Position: Topological Deep Learning is the New Frontier for Relational Learning. In R. Salakhutdinov, Z. Kolter, K. Heller, A. Weller, N. Oliver, J. Scarlett, & F. Berkenkamp (Eds.), Proceedings of the 41st International Conference on Machine Learning (Vol. 235, pp. 39529–39555). PMLR. https://proceedings.mlr.press/v235/papamarkou24a.html
    [ PDF ]
  • Neuhäuser, L., Scholkemper, M., Tudisco, F., & Schaub, M. T. (2024). Learning the effective order of a hypergraph dynamical system. Science Advances, 10(19). https://doi.org/10.1126/sciadv.adh4053
  • Grande, V. P., & Schaub, M. T. (2024). Disentangling the Spectral Properties of the Hodge Laplacian: not all small Eigenvalues are Equal. ICASSP 2024 - 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 9896–9900. https://doi.org/10.1109/icassp48485.2024.10446051
  • Hoppe, J., & Schaub, M. T. (2024). Representing Edge Flows on Graphs via Sparse Cell Complexes. In S. Villar & B. Chamberlain (Eds.), Proceedings of the Second Learning on Graphs Conference (Vol. 231, p. 1:1-1:22). PMLR. https://proceedings.mlr.press/v231/hoppe24a.html
    [ PDF ] [ Poster ] [ Slides ] [ Code ] [ PyPI ]
  • Grande, V. P., & Schaub, M. T. (2024). Non-Isotropic Persistent Homology: Leveraging the Metric Dependency of PH. In S. Villar & B. Chamberlain (Eds.), Proceedings of the Second Learning on Graphs Conference (Vol. 231, p. 17:1-17:19). PMLR. https://proceedings.mlr.press/v231/grande24a.html
    [ PDF ]
  • Grande, V. P., Hoppe, J., Frantzen, F., & Schaub, M. T. (2024). Topological Trajectory Classification and Landmark Inference on Simplicial Complexes. 2024 58th Asilomar Conference on Signals, Systems, and Computers, 44–48. https://doi.org/10.1109/ieeeconf60004.2024.10942887
  • Epping, B., René, A., Helias, M., & Schaub, M. T. (2024). Graph Neural Networks Do Not Always Oversmooth. In A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, & C. Zhang (Eds.), Advances in Neural Information Processing Systems (Vol. 37, pp. 48164–48188). Curran Associates, Inc.
    [ PDF ] [ Code ]
  • Frantzen, F., & Schaub, M. T. (2024). Learning From Simplicial Data Based on Random Walks and 1D Convolutions. The Twelfth International Conference on Learning Representations.
    [ PDF ] [ Code ]
  • Hajij, M., Papillon, M., Frantzen, F., Agerberg, J., AlJabea, I., Ballester, R., Battiloro, C., Bernárdez, G., Birdal, T., Brent, A., Chin, P., Escalera, S., Fiorellino, S., Gardaa, O. H., Gopalakrishnan, G., Govil, D., Hoppe, J., Karri, M. R., Khouja, J., … Miolane, N. (2024). TopoX: A Suite of Python Packages for Machine Learning on Topological Domains. Journal of Machine Learning Research, 25(374), 1–8. http://jmlr.org/papers/v25/24-0110.html
    [ PDF ] [ GitHub ]

2023

  • Grande, V. P., & Schaub, M. T. (2023). Topological Point Cloud Clustering. In A. Krause, E. Brunskill, K. Cho, B. Engelhardt, S. Sabato, & J. Scarlett (Eds.), Proceedings of the 40th International Conference on Machine Learning (Vol. 202, pp. 11683–11697). PMLR. https://proceedings.mlr.press/v202/grande23a.html
    [ PDF ]
  • Hajij, M., Zamzmi, G., Papamarkou, T., Guzman-Saenz, Ai., Birdal, T., & Schaub, M. T. (2023). Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs. 2023 57th Asilomar Conference on Signals, Systems, and Computers, 799–803. https://doi.org/10.1109/ieeeconf59524.2023.10477018