Graph Neural Networks excel at learning from pairwise interactions but often overlook multi-way and hierarchical relationships. Topological Deep Learning addresses this limitation by leveraging combinatorial topological spaces. However, existing TDL models are restricted to undirected settings and fail to capture the higher-order directed patterns prevalent in many complex systems, e.g., brain networks, where such interactions are both abundant and functionally significant. To fill this gap, we introduce Semi-Simplicial Neural Networks (SSNs), a principled class of TDL models that operate on semi-simplicial sets — combinatorial structures that encode directed higher-order motifs and their directional relationships. To enhance scalability, we propose Routing-SSNs, which dynamically select the most informative relations in a learnable manner. SSNs achieve state-of-the-art performance on brain dynamics classification tasks, outperforming the second-best model by up to 27%, and message-passing GNNs by up to 50% in accuracy.
@inproceedings{lecha2026directed,title={Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding},author={Lecha, Manuel and Cavallo, Andrea and Dominici, Francesca and Levi, Ran and Del Bue, Alessio and Isufi, Elvin and Morerio, Pietro and Battiloro, Claudio},booktitle={International Conference on Learning Representations (ICLR)},year={2026},url={https://openreview.net/forum?id=YR3CNvFfCr}}
2025
ICASSP
Higher-Order Topological Directionality and Directed Simplicial Neural Networks
Manuel Lecha, Andrea Cavallo, Francesca Dominici, and 2 more authors
In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2025
We introduce a notion of higher-order directionality and design Directed Simplicial Neural Networks (Dir-SNNs), message-passing networks operating on directed simplicial complexes that leverage directed interactions among simplices. To our knowledge, this is the first TDL model using a notion of higher-order directionality. We prove that Dir-SNNs are strictly more expressive than their directed graph counterparts in distinguishing isomorphic directed graphs, and show empirically that they outperform undirected SNNs when the underlying complex is directed.
@inproceedings{lecha2025higherorder,title={Higher-Order Topological Directionality and Directed Simplicial Neural Networks},author={Lecha, Manuel and Cavallo, Andrea and Dominici, Francesca and Isufi, Elvin and Battiloro, Claudio},booktitle={IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)},year={2025},}
DMLR
TopoBench: A Framework for Benchmarking Topological Deep Learning
Lev Telyatnikov, Guillermo Bernárdez, Marco Montagna, and 4 more authors
Data-centric Machine Learning Research (DMLR), 2025
@article{telyatnikov2025topobench,title={{TopoBench}: A Framework for Benchmarking Topological Deep Learning},author={Telyatnikov, Lev and Bernárdez, Guillermo and Montagna, Marco and Hajij, Mustafa and Ferriol Galmés, Miquel and Lecha, Manuel and others},journal={Data-centric Machine Learning Research (DMLR)},year={2025},}
JMLR
TopoX: A Suite of Python Packages for Machine Learning on Topological Domains
Mustafa Hajij, Mathilde Papillon, Florian Frantzen, and 4 more authors
@article{hajij2025topox,title={{TopoX}: A Suite of Python Packages for Machine Learning on Topological Domains},author={Hajij, Mustafa and Papillon, Mathilde and Frantzen, Florian and Agerberg, Jens and AlJabea, Ibrahem and Ballester, Rubén and others},journal={Journal of Machine Learning Research (JMLR)},year={2025},}
TAG-DS
Topological Deep Learning Challenge 2025: Expanding the Data Landscape
Guillermo Bernárdez, Lev Telyatnikov, Mathilde Papillon, and 4 more authors
In Topology, Algebra, and Geometry in Data Science Workshop (TAG-DS), 2025
@inproceedings{bernardez2025tdl2025,title={Topological Deep Learning Challenge 2025: Expanding the Data Landscape},author={Bernárdez, Guillermo and Telyatnikov, Lev and Papillon, Mathilde and Montagna, Marco and Theiler, Raphaël and Lecha, Manuel and others},booktitle={Topology, Algebra, and Geometry in Data Science Workshop (TAG-DS)},year={2025}}
arXiv
E-M3RF: An Equivariant Multimodal 3D Re-assembly Framework
Adeela Islam, Stefano Fiorini, Manuel Lecha, and 4 more authors
@misc{islam2025em3rf,title={{E-M3RF}: An Equivariant Multimodal 3D Re-assembly Framework},author={Islam, Adeela and Fiorini, Stefano and Lecha, Manuel and Tsesmelis, Theodore and James, Stuart and Morerio, Pietro and Del Bue, Alessio},year={2025},note={arXiv preprint}}
2024
ICML-W
ICML Topological Deep Learning Challenge 2024: Beyond the Graph Domain
Guillermo Bernárdez, Lev Telyatnikov, Marco Montagna, and 4 more authors
In ICML 2024 Workshop on Geometry-grounded Representation Learning and Generative Modeling (GRaM), 2024
@inproceedings{bernardez2024tdl2024,title={{ICML} Topological Deep Learning Challenge 2024: Beyond the Graph Domain},author={Bernárdez, Guillermo and Telyatnikov, Lev and Montagna, Marco and Baccini, Federica and Papillon, Mathilde and Lecha, Manuel and others},booktitle={ICML 2024 Workshop on Geometry-grounded Representation Learning and Generative Modeling (GRaM)},year={2024},}
2023
PMLR
ICML 2023 Topological Deep Learning Challenge: Design and Results
Mathilde Papillon, Mustafa Hajij, Audun Myers, and 5 more authors
In Topological, Algebraic and Geometric Learning Workshops, 2023
@inproceedings{papillon2023tdl2023,title={{ICML} 2023 Topological Deep Learning Challenge: Design and Results},author={Papillon, Mathilde and Hajij, Mustafa and Myers, Audun and Jenne, Helen and Mathe, Johan and Papamarkou, Theodore and Lecha, Manuel and others},booktitle={Topological, Algebraic and Geometric Learning Workshops},year={2023},url={https://proceedings.mlr.press/v221/papillon23a/papillon23a.pdf}}