[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118640-en":3,"doc-seo-118640-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},118640,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Group invariant machine learning by fundamental domain projections - Abstract","Supervised group-invariant and group-equivariant machine learning is studied through geometric topology. The approach introduces a pre-processing projection that maps inputs into a geometric space parameterizing symmetry-group orbits. The projected data can then be fed into any learning model, including neural networks, random forests, or support-vector machines. An efficient algorithm is provided for computing the geometric projection, and experiments show improved accuracy on example problems such as predicting Hodge numbers of CICY matrices.","King’s Research Portal  \nDocument Version  \nPublisher's PDF, also known as Version of record  \nLink to publication record in King's Research Portal  \nCitation for published version (APA):  \nSheard, D. , Platt, D. , & Aslan, B. (2023) . Group invariant machine learning by fundamental domain projections. Proceedings of Machine Learning Research, 197, 182-218 . [https://proceedings.mlr.press/v197/aslan23a.html](https://proceedings.mlr.press/v197/aslan23a.html)  \nCiting this paper  \nPlease note that where the full-text provided on King's Research Portal is the Author Accepted Manuscript or Post-Print version this may differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. 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Dec. 2025  \nGroup invariant machine learning by fundamental domain projections  \nBenjamin Aslan∗ [ucahbas@ucl.ac.uk](ucahbas@ucl.ac.uk)  \nUniversity College London  \nDaniel Platt∗ [daniel.1.platt@kcl.ac.uk](daniel.1.platt@kcl.ac.uk)  \nKing’s College London, Strand Campus  \nDavid Sheard∗ [david.sheard.17@ucl.ac.uk](david.sheard.17@ucl.ac.uk)  \nUniversity College London  \nEditors: Sophia Sanborn, Christian Shewmake, Simone Azeglio, Arianna Di Bernardo, Nina Miolane  \nAbstract  \nWe approach the well-studied problem of supervised group invariant and equivariant machine learning from the point of view of geometric topology. We propose a novel approach using a pre-processing step, which involves projecting the input data into a geometric space which parametrises the orbits of the symmetry group. This new data can then be the input for an arbitrary machine learning model (neural network, random forest, support-vector machine etc) . We give an algorithm to compute the geometric projection, which is efficient to implement, and we illustrate our approach on some example machine learning problems (including the well-studied problem of predicting Hodge numbers of CICY matrices), finding an improvement in accuracy versus others in the literature.  \nKeywords: Group invariant, group equivariant, geometric deep learning, fundamental domain, geometric topology  \n1. Introduction  \nMany tasks in machine learning can be understood as approximating a function α : X → Y between a feature space and an output space. Typically, these may be subsets of Rn, but could be more complicated like Riemannian manifolds. We consider the problem in the presence of symmetries—more precisely, suppose a group G that acts on X on the left, and α satisfies the invariance property  \nα (g · x) = α(x) for all x ∈ X, g ∈ G. (1)  \nA simple example is recognising a single handwritten digit which may have been rotated by 90◦ , 180◦ , or 270◦ , so the problem is invariant under the action of Z4 .  \nMachine learning models such as neural networks or random forests can approximate α but the resulting function β will not generally be G-invariant. The key t","cbCaimX7pr3v9053","https://ap.wps.com/l/cbCaimX7pr3v9053","pdf",883454,1,38,"English","en",105,"# Abstract\n# 1. Introduction\n## 1.1. Previous work","[{\"question\":\"How does the method ensure group invariance in machine learning?\",\"answer\":\"It uses a pre-processing projection that maps inputs to a geometric space parameterizing the orbits of the symmetry group, enabling invariant learning by construction with standard models.\"},{\"question\":\"What is the role of geometric topology in the proposed approach?\",\"answer\":\"Geometric topology provides the viewpoint and structure for projecting data into an orbit-parameterizing space, linking symmetries to the geometry used for learning.\"},{\"question\":\"What kinds of machine learning models can use the projected data?\",\"answer\":\"Any arbitrary machine learning model can be applied to the projected data, including neural networks, random forests, and support-vector machines.\"}]","Group invariant machine learning by fundamental domain projections - Abstract | PDF",1785684651,96,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"group-invariant-machine-learning-by-fundamental-domain-projections-abstract","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/group-invariant-machine-learning-by-fundamental-domain-projections-abstract/118640/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How does the method ensure group invariance in machine learning?","Question",{"text":75,"@type":76},"It uses a pre-processing projection that maps inputs to a geometric space parameterizing the orbits of the symmetry group, enabling invariant learning by construction with standard models.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the role of geometric topology in the proposed approach?",{"text":80,"@type":76},"Geometric topology provides the viewpoint and structure for projecting data into an orbit-parameterizing space, linking symmetries to the geometry used for learning.",{"name":82,"@type":73,"acceptedAnswer":83},"What kinds of machine learning models can use the projected data?",{"text":84,"@type":76},"Any arbitrary machine learning model can be applied to the projected data, including neural networks, random forests, and support-vector machines.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]