{"work":{"id":"5e909969-dcfb-40f6-9099-241ea6f18350","openalex_id":"https://openalex.org/W3157286395","doi":"10.48550/arxiv.2104.13478","arxiv_id":"2104.13478","raw_key":null,"title":"Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges","authors":null,"authors_text":"Michael M. Bronstein, Joan Bruna, Taco Cohen, Petar Veli\\v{c}kovi\\'c","year":2021,"venue":"cs.LG","abstract":"The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation.\n  While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications.\n  Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.","external_url":"https://arxiv.org/abs/2104.13478","cited_by_count":557,"metadata_source":"pith","metadata_fetched_at":"2026-08-05T02:28:24.338817+00:00","pith_arxiv_id":"2104.13478","created_at":"2026-05-09T06:20:37.684491+00:00","updated_at":"2026-08-05T02:28:24.338817+00:00","title_quality_ok":true,"display_title":"Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges","render_title":"Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges"},"hub":{"state":{"work_id":"5e909969-dcfb-40f6-9099-241ea6f18350","tier":"super_hub","tier_reason":"100+ Pith inbound or 10,000+ external citations","pith_inbound_count":122,"external_cited_by_count":557,"distinct_field_count":28,"first_pith_cited_at":"2023-01-04T18:15:07+00:00","last_pith_cited_at":"2026-07-09T04:01:25+00:00","author_build_status":"needed","summary_status":"needed","contexts_status":"needed","graph_status":"needed","ask_index_status":"needed","reader_status":"not_needed","recognition_status":"not_needed","updated_at":"2026-08-22T14:19:24.154422+00:00","tier_text":"super_hub"},"tier":"super_hub","role_counts":[{"context_role":"background","n":15}],"polarity_counts":[{"context_polarity":"background","n":15}],"runs":{"ask_index":{"job_type":"ask_index","status":"succeeded","result":{"title":"Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges","claims":[{"claim_text":"The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second,","claim_type":"abstract","evidence_strength":"source_metadata"},{"claim_text":"06128, 2018. [11] Tarek R Besold, Artur d'Avila Garcez, Sebastian Bader, Howard Bowman, Pedro Domingos, Pascal Hitzler, Kai-Uwe Levy, Luis C Lamb, et al. Neural-symbolic learning and reasoning: A survey and interpretation.arXiv preprint arXiv:1711.03902, 2017. [12] Stephen Boyd and Lieven Vandenberghe.Convex optimization. Cambridge University Press, 2004. [13] Michael M Bronstein, Joan Bruna, Taco Cohen, and Petar Veliˇckovi'c. 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InProceedings of the 40th International Conference on Machine Learning, volume 202 ofProceedings of Machine Learning Research, pages 26517-26582, 2023. [58] Rong Tang and Yun Yang. Adaptivity of diffusion models to manifold structures. InInternational Conference on Artificial Intelligence and Statist","claim_type":"background","confidence":0.9,"evidence_strength":"citation_context"},{"claim_text":"be the more fundamental object and construct a graph through it, we forgo any mention to permutations of the nodes and simply call the graph circulant if the adjacency matrix we use for its construction is circulant. 2 can be defined in a natural way [3]. 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InProceedings of the 40th International Conference on Machine Learning, volume 202 ofProceedings of Machine Learning Research, pages 26517-26582, 2023. [58] Rong Tang and Yun Yang. Adaptivity of diffusion models to manifold structures. InInternational Conference on Artificial Intelligence and Statist","claim_type":"background","confidence":0.9,"evidence_strength":"citation_context"},{"claim_text":"be the more fundamental object and construct a graph through it, we forgo any mention to permutations of the nodes and simply call the graph circulant if the adjacency matrix we use for its construction is circulant. 2 can be defined in a natural way [3]. 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Such graphs have seen increased use in the field of machine learning with the rise of Geometric Deep Learning [4], where their inherent rotational invariance is leveraged [5], as well as a means to compress the weight matrices' sizes and expedi","claim_type":"background","confidence":0.85,"evidence_strength":"citation_context"},{"claim_text":"InInternational conference on machine learning, pp. 2806-2823. PMLR, 2023. [20] Viacheslav Borovitskiy, Alexander Terenin, Peter Mostowsky, and Marc Peter Deisenroth. Matérn gaussian processes on Riemannian manifolds. InAdvances in Neural Information Processing Systems, volume 33, 2020. URL https://proceedings.neurips. cc/paper/2020/hash/92bf5e6240737e0326ea59846a83e076-Abstract. html. [21] Michael M Bronstein, Joan Bruna, Taco Cohen, and Petar Veliˇckovi'c. Geometric deep learning: Grids, group","claim_type":"background","confidence":0.8,"evidence_strength":"citation_context"},{"claim_text":"terpreted as the valuef θ(x) of the model. A growing body of literature argues that Quantum Neural Networks have a spectral bias [39-42] which can be manipulated for specific learning tasks [95]. This includes a \"hard\" spec- tral bias stemming from the embedding of classical data and a potential \"soft\" spectral bias that regularises the underlying model class [39], as well as a possible spectral bias with respect to the learning dynamics similar to the one observed in classical neural networks [","claim_type":"background","confidence":0.8,"evidence_strength":"citation_context"}],"why_cited":"Pith tracks Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges because it crossed a citation-hub threshold. 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