Publications
Papers
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2026
Intrinsic Geometry Matching: Learning Observation-Equivariant Process Geometry
Real data are often arbitrary, indirect observations of an unknown underlying process, and those observations can distort its geometry. Existing geometric methods equip data with the induced geometry of the ambient space, but this entangles the true geometry of the process with the distortion of the observation. To address this, we introduce Intrinsic Geometry Matching (IGM): a scalable approach to learn the intrinsic geometry of the underlying process. Specifically, we learn the carré du champ operator of the process, allowing us to recover its diffusion geometry. Our method uses the fact that data often come with natural perturbations, like steps in a time series, to recover the intrinsic carré du champ while transforming naturally with changes in observation: it is observation-equivariant. Furthermore, since this equivariance relates the geometry of different spaces, we can train a model using additional sources of data, even when that data lies in another space. We evaluate IGM on molecular dynamics, image, and video data and find better performance and scaling than existing neural and graph-based geometric tools.
@misc{lecha2026igm, author = {Lecha, Manuel and Bronstein, Michael and Jones, Iolo}, title = {Intrinsic Geometry Matching: Learning Observation-Equivariant Process Geometry}, year = {2026}, eprint = {2610.16995}, archiveprefix = {arXiv}, howpublished = {arXiv preprint arXiv:2610.16995}, primaryclass = {cs.LG} }Expressivity Saturation: Reduced Affine Region Usage Under Increasing Task Complexity
- paper: Expressivity Saturation: Reduced Affine Region Usage Under Increasing Task Complexity
- arXiv: Expressivity Saturation: Reduced Affine Region Usage Under Increasing Task Complexity
Piecewise-affine neural networks (e.g., with ReLU or LeakyReLU activations) implement continuous piecewise-affine maps, and the number of affine regions provides a natural proxy for expressive capacity. However, the gap between theoretical region capacity and the affine regions realized after training remains insufficiently understood. We study this gap from two complementary perspectives. First, we give a rigorous, architecture-dependent theorem for affine line-segment probes: for multilayer perceptrons with piecewise-affine activations, the number of affine pieces realized along an affine line-segment probe is upper bounded by an explicit product of layer-wise width terms (and activation breakpoint factors). This yields a neuron-threshold lower bound for representing target functions with prescribed one-dimensional piece complexity, formalizing the minimal region budget required for complex signals. Second, we exactly enumerate affine regions realized within bounded 2D and higher-dimensional domains under controlled task complexity. Under fixed architectures and training protocols, increasing input–label complexity yields trained solutions with markedly fewer realized regions in the evaluation domain, even though worst-case architectural capacity is unchanged; we call this reduced region usage expressivity saturation. Moreover, in the most challenging regimes, 2D visualizations show that region-usage collapse often coincides with degraded decision boundaries. Finally, we visualize the training dynamics of affine-region partitions and decision boundaries, revealing a consistent refinement process during optimization.
@article{qi2026expressivity, author = {Qi, Xuan and Wei, Yi and Yu, Fanqi and Lecha, Manuel}, title = {Expressivity Saturation: Reduced Affine Region Usage Under Increasing Task Complexity}, journal = {Transactions on Machine Learning Research (TMLR)}, year = {2026}, url = {https://openreview.net/forum?id=JiyZE3yKv8}, eprint = {2606.21687}, archiveprefix = {arXiv} }E-M3RF: An Equivariant Multimodal 3D Re-assembly Framework
- paper: E-M3RF: An Equivariant Multimodal 3D Re-assembly Framework
- arXiv: E-M3RF: An Equivariant Multimodal 3D Re-assembly Framework
- code: E-M3RF: An Equivariant Multimodal 3D Re-assembly Framework
3D reassembly is a fundamental geometric problem, and in recent years it has increasingly been challenged by deep learning methods rather than classical optimization. While learning approaches have shown promising results, most still rely primarily on geometric features to assemble a whole from its parts. This reliance limits their effectiveness in cases where geometric information is incomplete or degraded, such as for small or eroded fragments. Additionally, solutions do not impose physical constraints that explicitly prevent overlapping assemblies. To address these limitations, we introduce E-M3RF, an equivariant multimodal 3D reassembly framework that takes as input the point clouds, containing both point positions and color of fractured fragments, and predicts the transformations required to reassemble them, using SE(3) flow matching. Each fragment is represented by both geometric and color features: i) 3D point positions are encoded as rotation-consistent geometric features using a rotation-equivariant encoder, while ii) the colors at each 3D point are encoded with a transformer. The two feature sets are then combined to form a multimodal representation. We experimented on five datasets: two synthetic datasets, Breaking Bad and Fantastic Breaks, and three real-world cultural heritage datasets, RePAIR, Presious, and FRACTURA, demonstrating that E-M3RF on the RePAIR dataset (the most challenging one) reduces rotation error by 23.1% and translation error by 13.2%, while Chamfer Distance decreases by 18.4% compared to competing methods.
@inproceedings{islam2026em3rf, author = {Islam, Adeela and Fiorini, Stefano and Lecha, Manuel and Tsesmelis, Theodore and James, Stuart and Morerio, Pietro and Del Bue, Alessio}, title = {{E-M3RF}: An Equivariant Multimodal {3D} Re-assembly Framework}, booktitle = {European Conference on Computer Vision (ECCV)}, series = {Lecture Notes in Computer Science}, volume = {17060}, pages = {683--700}, year = {2026}, publisher = {Springer Nature Switzerland}, doi = {10.1007/978-3-032-37092-1_38}, url = {https://link.springer.com/chapter/10.1007/978-3-032-37092-1_38}, eprint = {2511.21422}, archiveprefix = {arXiv} }Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding
- paper: Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding
- arXiv: Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding
- code: Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding
Graph Neural Networks (GNNs) excel at learning from pairwise interactions but often overlook multi-way and hierarchical relationships. Topological Deep Learning (TDL) 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. We prove that SSNs are strictly more expressive than standard graph and TDL models. We then introduce a new principled framework for brain dynamics representation learning, grounded in the ability of SSNs to provably recover topological descriptors shown to successfully characterize brain activity. Empirically, 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. Our results highlight the potential of principled topological models for learning from structured brain data, establishing a unique real-world case study for TDL. We also test SSNs on standard node classification and edge regression tasks, showing competitive performance.
@inproceedings{lecha2026ssn, 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}, title = {Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding}, booktitle = {International Conference on Learning Representations (ICLR)}, pages = {81644--81690}, year = {2026}, url = {https://openreview.net/forum?id=YR3CNvFfCr}, eprint = {2505.17939}, archiveprefix = {arXiv} }Topological Deep Learning Challenge 2025: Expanding the Data Landscape
- paper: Topological Deep Learning Challenge 2025: Expanding the Data Landscape
- code: Topological Deep Learning Challenge 2025: Expanding the Data Landscape
This paper describes the 2025 edition of the Topological Deep Learning Challenge: Expanding the Data Landscape, hosted at the first Topology, Algebra, and Geometry in Data Science (TAG-DS) Conference. This year’s challenge aimed to address the data bottleneck in the field by systematically expanding the ecosystem of Topological Deep Learning (TDL). Powered by TopoBench, the challenge was organized into two primary missions: enriching the data landscape with diverse datasets, and advancing core data infrastructure. In particular, participants were invited to contribute to the open-source platform by implementing new dataset loaders, designing new benchmark tasks, or engineering robust, scalable data pipelines. The initiative successfully yielded 44 qualifying submissions. This paper outlines the scope of the competition and summarizes the key results and findings, highlighting the new resources now available to the TDL community.
@inproceedings{bernardez2025tdl, author = {Bernárdez, Guillermo and Telyatnikov, Lev and Papillon, Mathilde and Montagna, Marco and Theiler, Raffael and Cornelis, Louisa and Mathe, Johan and Ferriol, Miquel and Vasylenko, Pavlo and Van Looy, Jan-Willem and Testa, Lucia and Neri, Bruno and Genovese, Donatella and Weber, Melanie and Wei, Amaury and Devoto, Alessio and Weers, Alexander and Jankowski, Robert and Cino, Loris and Leko, David and Banf, Michael and Müller, Jonas and Grapentin, Thomas and Paik, Taejin and Dutta, Abhijeet and Walter, Hugo and Fontanari, Thomas Vaitses and Ghasemi, Ali and Loi, Dario and Sáez de Ocáriz Borde, Haitz and Aguilar-Argüello, Gabriela and da Rosa, Giovanni B. and Saulus, Théo and Dolores-Cuenca, Eric Rubiel and Di Nino, Leonardo and Leroy, Pierrick and Pandolfo, Mario Edoardo and Cavallo, Andrea and Qin, Yu and Snopov, Pavel and Akbari, Amirreza and Meza-Chávez, Ixchel and Van Langendonck, Louis and Able, Jared and Meshcheryakova, Maria Yuffa and Tsay, Henry and Benić, Luka and Filipiak, Dominik and Liu, Patrick and Liang, Huidong and Santos da Rosa Jr., Alexsandro and Cattai, Tiziana and Borges, Henrique M. and Grimaldi, Enrico and Lecha, Manuel and Battiloro, Claudio and Liu, Xuan-Chen and Deshpande, Raj and Johnson, Graham and Morgunov, Igor and Micheron, Hugo and Devaux, Rémi and Jardin, Antoine and Emerson, Tegan and Fink, Olga and Miolane, Nina}, title = {Topological Deep Learning Challenge 2025: Expanding the Data Landscape}, booktitle = {Proceedings of the 1st Conference on Topology, Algebra, and Geometry in Data Science (TAG-DS 2025)}, series = {Proceedings of Machine Learning Research}, volume = {321}, pages = {4--14}, year = {2026}, publisher = {PMLR}, url = {https://proceedings.mlr.press/v321/bernardez26a.html} }
2025
Higher-Order Topological Directionality and Directed Simplicial Neural Networks
- paper: Higher-Order Topological Directionality and Directed Simplicial Neural Networks
- arXiv: Higher-Order Topological Directionality and Directed Simplicial Neural Networks
- code: Higher-Order Topological Directionality and Directed Simplicial Neural Networks
Topological Deep Learning (TDL) has emerged as a paradigm to process and learn from signals defined on higher-order combinatorial topological spaces, such as simplicial or cell complexes. Although many complex systems have an asymmetric relational structure, most TDL models forcibly symmetrize these relationships. In this paper, we first introduce a novel notion of higher-order directionality and we then design Directed Simplicial Neural Networks (Dir-SNNs) based on it. Dir-SNNs are message-passing networks operating on directed simplicial complexes able to leverage directed and possibly asymmetric interactions among the simplices. To our knowledge, this is the first TDL model using a notion of higher-order directionality. We theoretically and empirically prove that Dir-SNNs are more expressive than their directed graph counterpart in distinguishing isomorphic directed graphs. Experiments on a synthetic source localization task demonstrate that Dir-SNNs outperform undirected SNNs when the underlying complex is directed, and perform comparably when the underlying complex is undirected.
@inproceedings{lecha2025dirsnn, author = {Lecha, Manuel and Cavallo, Andrea and Dominici, Francesca and Isufi, Elvin and Battiloro, Claudio}, title = {Higher-Order Topological Directionality and Directed Simplicial Neural Networks}, booktitle = {IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)}, pages = {1--5}, year = {2025}, publisher = {IEEE}, doi = {10.1109/ICASSP49660.2025.10890264}, url = {https://ieeexplore.ieee.org/document/10890264}, eprint = {2409.08389}, archiveprefix = {arXiv} }TopoBench: A Framework for Benchmarking Topological Deep Learning
- paper: TopoBench: A Framework for Benchmarking Topological Deep Learning
- arXiv: TopoBench: A Framework for Benchmarking Topological Deep Learning
- code: TopoBench: A Framework for Benchmarking Topological Deep Learning
This work introduces TopoBench, an open-source library designed to standardize benchmarking and accelerate research in topological deep learning (TDL). TopoBench decomposes TDL into a sequence of independent modules for data generation, loading, transforming and processing, as well as model training, optimization and evaluation. This modular organization provides flexibility for modifications and facilitates the adaptation and optimization of various TDL pipelines. A key feature of TopoBench is its support for transformations and lifting across topological domains. Mapping the topology and features of a graph to higher-order topological domains, such as simplicial and cell complexes, enables richer data representations and more fine-grained analyses. The applicability of TopoBench is demonstrated by benchmarking several TDL architectures across diverse tasks and datasets.
@article{telyatnikov2025topobench, author = {Telyatnikov, Lev and Bernárdez, Guillermo and Montagna, Marco and Hajij, Mustafa and Carrasco, Martin and Vasylenko, Pavlo and Papillon, Mathilde and Zamzmi, Ghada and Schaub, Michael T. and Verhellen, Jonas and Snopov, Pavel and Miquel-Oliver, Bertran and Gil-Sorribes, Manel and Molina, Alexis and Guallar, Victor and Long, Theodore and Suk, Julian and Rygiel, Patryk and Nikitin, Alexander and Escalona, Giordan and Banf, Michael and Filipiak, Dominik and Schattauer, Max and Imasheva, Liliya and Martinez, Alvaro and Fritze, Halley and Masden, Marissa and Sánchez, Valentina and Lecha, Manuel and Cavallo, Andrea and Battiloro, Claudio and Piekenbrock, Matt and Tec, Mauricio and Dasoulas, George and Miolane, Nina and Scardapane, Simone and Papamarkou, Theodore}, title = {{TopoBench}: A Framework for Benchmarking Topological Deep Learning}, journal = {Journal of Data-centric Machine Learning Research (DMLR)}, volume = {2}, number = {20}, pages = {1--39}, year = {2025}, url = {https://openreview.net/forum?id=07sTzyEVtY}, eprint = {2406.06642}, archiveprefix = {arXiv} }
2024
TopoX: A Suite of Python Packages for Machine Learning on Topological Domains
- paper: TopoX: A Suite of Python Packages for Machine Learning on Topological Domains
- arXiv: TopoX: A Suite of Python Packages for Machine Learning on Topological Domains
- code: TopoX: A Suite of Python Packages for Machine Learning on Topological Domains
We introduce TopoX, a Python software suite that provides reliable and user-friendly building blocks for computing and machine learning on topological domains that extend graphs: hypergraphs, simplicial, cellular, path and combinatorial complexes. TopoX consists of three packages: TopoNetX facilitates constructing and computing on these domains, including working with nodes, edges and higher-order cells; TopoEmbedX provides methods to embed topological domains into vector spaces, akin to popular graph-based embedding algorithms such as node2vec; TopoModelX is built on top of PyTorch and offers a comprehensive toolbox of higher-order message passing functions for neural networks on topological domains. The extensively documented and unit-tested source code of TopoX is available under MIT license at https://pyt-team.github.io.
@article{hajij2024topox, author = {Hajij, Mustafa and Papillon, Mathilde and Frantzen, Florian and Agerberg, Jens and AlJabea, Ibrahem and Ballester, Rubén and Battiloro, Claudio and Bernárdez, Guillermo and Birdal, Tolga and Brent, Aiden and Chin, Peter and Escalera, Sergio and Fiorellino, Simone and Gardaa, Odin Hoff and Gopalakrishnan, Gurusankar and Govil, Devendra and Hoppe, Josef and Karri, Maneel Reddy and Khouja, Jude and Lecha, Manuel and Livesay, Neal and Meißner, Jan and Mukherjee, Soham and Nikitin, Alexander and Papamarkou, Theodore and Prílepok, Jaro and Ramamurthy, Karthikeyan Natesan and Rosen, Paul and Guzmán-Sáenz, Aldo and Salatiello, Alessandro and Samaga, Shreyas N. and Scardapane, Simone and Schaub, Michael T. and Scofano, Luca and Spinelli, Indro and Telyatnikov, Lev and Truong, Quang and Walters, Robin and Yang, Maosheng and Zaghen, Olga and Zamzmi, Ghada and Zia, Ali and Miolane, Nina}, title = {{TopoX}: A Suite of {Python} Packages for Machine Learning on Topological Domains}, journal = {Journal of Machine Learning Research (JMLR)}, volume = {25}, number = {374}, pages = {1--8}, year = {2024}, url = {https://jmlr.org/papers/v25/24-0110.html}, eprint = {2402.02441}, archiveprefix = {arXiv} }ICML Topological Deep Learning Challenge 2024: Beyond the Graph Domain
- paper: ICML Topological Deep Learning Challenge 2024: Beyond the Graph Domain
- arXiv: ICML Topological Deep Learning Challenge 2024: Beyond the Graph Domain
- code: ICML Topological Deep Learning Challenge 2024: Beyond the Graph Domain
This paper describes the 2nd edition of the ICML Topological Deep Learning Challenge that was hosted within the ICML 2024 ELLIS Workshop on Geometry-grounded Representation Learning and Generative Modeling (GRaM). The challenge focused on the problem of representing data in different discrete topological domains in order to bridge the gap between Topological Deep Learning (TDL) and other types of structured datasets (e.g. point clouds, graphs). Specifically, participants were asked to design and implement topological liftings, i.e. mappings between different data structures and topological domains –like hypergraphs, or simplicial/cell/combinatorial complexes. The challenge received 52 submissions satisfying all the requirements. This paper introduces the main scope of the challenge, and summarizes the main results and findings.
@inproceedings{bernardez2024tdl, author = {Bernárdez, Guillermo and Telyatnikov, Lev and Montagna, Marco and Baccini, Federica and Papillon, Mathilde and Ferriol-Galmés, Miquel and Hajij, Mustafa and Papamarkou, Theodore and Bucarelli, Maria Sofia and Zaghen, Olga and Mathe, Johan and Myers, Audun and Mahan, Scott and Lillemark, Hansen and Vadgama, Sharvaree and Bekkers, Erik and Doster, Tim and Emerson, Tegan and Kvinge, Henry and Agate, Katrina and Ahmed, Nesreen K and Bai, Pengfei and Banf, Michael and Battiloro, Claudio and Beketov, Maxim and Bogdan, Paul and Carrasco, Martin and Cavallo, Andrea and Choi, Yun Young and Dasoulas, George and Elphick, Matouš and Escalona, Giordan and Filipiak, Dominik and Fritze, Halley and Gebhart, Thomas and Gil-Sorribes, Manel and Goomanee, Salvish and Guallar, Victor and Imasheva, Liliya and Irimia, Andrei and Jin, Hongwei and Johnson, Graham and Kanakaris, Nikos and Koloski, Boshko and Kovač, Veljko and Lecha, Manuel and Lee, Minho and Leroy, Pierrick and Long, Theodore and Magai, German and Martinez, Alvaro and Masden, Marissa and Mežnar, Sebastian and Miquel-Oliver, Bertran and Molina, Alexis and Nikitin, Alexander and Nurisso, Marco and Piekenbrock, Matt and Qin, Yu and Rygiel, Patryk and Salatiello, Alessandro and Schattauer, Max and Snopov, Pavel and Suk, Julian and Sánchez, Valentina and Tec, Mauricio and Vaccarino, Francesco and Verhellen, Jonas and Wantiez, Frederic and Weers, Alexander and Zajec, Patrik and Škrlj, Blaž and Miolane, Nina}, title = {{ICML} Topological Deep Learning Challenge 2024: Beyond the Graph Domain}, booktitle = {Proceedings of the Geometry-grounded Representation Learning and Generative Modeling Workshop (GRaM)}, series = {Proceedings of Machine Learning Research}, volume = {251}, pages = {420--428}, year = {2024}, publisher = {PMLR}, url = {https://proceedings.mlr.press/v251/bernardez24a.html}, eprint = {2409.05211}, archiveprefix = {arXiv} }
2023
ICML 2023 Topological Deep Learning Challenge: Design and Results
- paper: ICML 2023 Topological Deep Learning Challenge: Design and Results
- arXiv: ICML 2023 Topological Deep Learning Challenge: Design and Results
- code: ICML 2023 Topological Deep Learning Challenge: Design and Results
This paper presents the computational challenge on topological deep learning that was hosted within the ICML 2023 Workshop on Topology and Geometry in Machine Learning. The competition asked participants to provide open-source implementations of topological neural networks from the literature by contributing to the python packages TopoNetX (data processing) and TopoModelX (deep learning). The challenge attracted twenty-eight qualifying submissions in its two-month duration. This paper describes the design of the challenge and summarizes its main findings.
@inproceedings{papillon2023tdl, author = {Papillon, Mathilde and Hajij, Mustafa and Jenne, Helen and Mathe, Johan and Myers, Audun and Papamarkou, Theodore and Birdal, Tolga and Dey, Tamal and Doster, Tim and Emerson, Tegan and Gopalakrishnan, Gurusankar and Govil, Devendra and Guzmán-Sáenz, Aldo and Kvinge, Henry and Livesay, Neal and Mukherjee, Soham and Samaga, Shreyas N. and Ramamurthy, Karthikeyan Natesan and Karri, Maneel Reddy and Rosen, Paul and Sanborn, Sophia and Walters, Robin and Agerberg, Jens and Barikbin, Sadrodin and Battiloro, Claudio and Bazhenov, Gleb and Bernardez, Guillermo and Brent, Aiden and Escalera, Sergio and Fiorellino, Simone and Gavrilev, Dmitrii and Hassanin, Mohammed and Häusner, Paul and Gardaa, Odin Hoff and Khamis, Abdelwahed and Lecha, Manuel and Magai, German and Malygina, Tatiana and Ballester, Rubén and Nadimpalli, Kalyan and Nikitin, Alexander and Rabinowitz, Abraham and Salatiello, Alessandro and Scardapane, Simone and Scofano, Luca and Singh, Suraj and Sjölund, Jens and Snopov, Pavel and Spinelli, Indro and Telyatnikov, Lev and Testa, Lucia and Yang, Maosheng and Yue, Yixiao and Zaghen, Olga and Zia, Ali and Miolane, Nina}, title = {{ICML} 2023 Topological Deep Learning Challenge: Design and Results}, booktitle = {Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML)}, series = {Proceedings of Machine Learning Research}, volume = {221}, pages = {3--8}, year = {2023}, publisher = {PMLR}, url = {https://proceedings.mlr.press/v221/papillon23a.html}, eprint = {2309.15188}, archiveprefix = {arXiv} }