[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121887-en":3,"doc-seo-121887-105":30,"detail-sidebar-cat-0-en-105":95},{"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},121887,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Normed Spaces for Graph Embedding - Abstract","The paper studies how normed spaces can embed finite metric spaces and, by extension, provide an efficient geometric foundation for learning graph embeddings. Theoretical insights from discrete geometry indicate low distortion bounds in low dimensions, motivating normed spaces—especially l1 and l_d—instead of many Riemannian manifolds. Experiments show superior graph reconstruction performance, robustness across negative/zero/positive curvature families, improved scalability, and strong results on link prediction and recommender systems while using fewer computational resources.","Normed Spaces for Graph Embedding  \nDiaaeldin Taha∗ [diaaeldin.taha@mis.mpg. de](diaaeldin.taha@mis.mpg. de)  \nMax Planck Institute for Mathematics in the Sciences, Leipzig, Germany  \nWei Zhao∗ [wei.zhao@abdn. ac.uk](wei.zhao@abdn. ac.uk)  \nUniversity of Aberdeen, Aberdeen, United Kingdom  \nJ. Maxwell Riestenberg [max.riestenberg@mis.mpg. de](max.riestenberg@mis.mpg. de)[ ](max.riestenberg@mis.mpg. de)Max Planck Institute for Mathematics in the Sciences,  \nLeipzig, Germany  \nMichael Strube [michael.strube@h-its. org](michael.strube@h-its. org)  \nHeidelberg Institute for Theoretical Studies, Heidelberg, Germany  \nReviewed on OpenReview: [https: // openreview. net/ forum? id= 4E2XLydJiv](https: // openreview. net/ forum? id= 4E2XLydJiv)  \nAbstract  \nTheoretical results from discrete geometry suggest that normed spaces can abstractly embed finite metric spaces with surprisingly low theoretical bounds on distortion in low dimensions.  \nInspired by this theoretical insight, we highlight in this paper normed spaces as a more flexible and computationally efficient alternative to several popular Riemannian manifolds for learning graph embeddings. Normed space embeddings significantly outperform several popular manifolds on a large range of synthetic and real-world graph reconstruction benchmark datasets while requiring significantly fewer computational resources. We also empirically verify the superiority of normed space embeddings on growing families of graphs associated with negative, zero, and positive curvature, further reinforcing the flexibility of normed spaces in capturing diverse graph structures as graph sizes increase. Lastly, we demonstrate the utility of normed space embeddings on two applied graph embedding tasks, namely, link prediction and recommender systems. Our work highlights the potential of normed spaces for geometric graph representation learning, raises new research questions, and offers a valuable tool for experimental mathematics in the field of finite metric space embeddings. We make our code and data publically available 1 .  \n1 Introduction  \nGraph representation learning aims to embed real-world graph data into ambient spaces while sufficiently preserving the geometric and statistical graph structures for subsequent downstream tasks and analysis. Graph data in many domains exhibit non-Euclidean features, making Euclidean embedding spaces an unfit choice. Motivated by the manifold hypothesis (see, e.g., Bengio et al. (2013)), recent research work has proposed embedding graphs into Riemannian manifolds (Chamberlain et al., 2017; Defferrard et al., 2020; Grattarola et al., 2020; Gu et al., 2019; Tifrea et al., 2019) . These manifolds introduce inductive biases, such as symmetry and curvature, that can match the underlying graph properties, thereby enhancing the quality of the embeddings. For instance, Chamberlain et al. (2017) and Defferrard et al. (2020) proposed embedding graphs into hyperbolic and spherical spaces, with the choice determined by the graph structures.  \n1[https://github.com/andyweizhao/graphs-normed-spaces](https://github.com/andyweizhao/graphs-normed-spaces)  \n∗ These authors contributed equally to this work.  \nMore recently, López et al. (López et al., 2021; López et al., 2021) proposed Riemannian symmetric spaces asa framework that unifies many Riemannian manifolds previously considered for representation learning. They also highlighted the Siegel and SPD symmetric spaces, whose geometries combine the sought-for inductive biases of many manifolds. However, operations in these non-Euclidean spaces are computationally demanding and technically challenging, making them impractical for embedding large graphs.  \nIn this work, we highlight normed spaces, particularly ℓd1 and ℓ, as a more flexible, more computationally efficient, and less technically challenging alternative to several popular Riemannian manifolds for learning graph embeddings. In particular, normed spaces are empirically observ","cbCaipGov2GMCBIu","https://ap.wps.com/l/cbCaipGov2GMCBIu","pdf",1183461,1,27,"English","en",105,"# Abstract\n# Introduction\n# Related Work","[{\"question\":\"Why do normed spaces help with graph embedding compared to Euclidean and Riemannian manifolds?\",\"answer\":\"Normed spaces are motivated by discrete geometry results showing low theoretical distortion bounds in low dimensions. The paper also reports easier training and better performance in practical graph embedding settings that use gradient descent.\"},{\"question\":\"How do normed space embeddings perform on graph reconstruction benchmarks?\",\"answer\":\"The work evaluates normed space capacity via a graph reconstruction task and finds that normed spaces outperform several popular manifold families across a broad range of synthetic and real-world datasets.\"},{\"question\":\"Does the advantage of normed spaces depend on graph curvature or size?\",\"answer\":\"Empirical results confirm superiority for growing graph families associated with negative, zero, and positive curvature, and show robustness as graph sizes increase. Computational resource growth is also slower than for alternative Riemannian approaches.\"},{\"question\":\"What applied tasks demonstrate the usefulness of normed space graph embeddings?\",\"answer\":\"The paper demonstrates utility on link prediction and recommender systems, with the l1 normed space surpassing baseline spaces in these applied graph embedding tasks.\"}]","Normed Spaces for Graph Embedding - Abstract | PDF",1785807502,68,{"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":90,"head_meta":92,"extra_data":94,"updated_unix":28},"normed-spaces-for-graph-embedding-abstract","",{"@graph":36,"@context":89},[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/normed-spaces-for-graph-embedding-abstract/121887/",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","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"Why do normed spaces help with graph embedding compared to Euclidean and Riemannian manifolds?","Question",{"text":75,"@type":76},"Normed spaces are motivated by discrete geometry results showing low theoretical distortion bounds in low dimensions. The paper also reports easier training and better performance in practical graph embedding settings that use gradient descent.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do normed space embeddings perform on graph reconstruction benchmarks?",{"text":80,"@type":76},"The work evaluates normed space capacity via a graph reconstruction task and finds that normed spaces outperform several popular manifold families across a broad range of synthetic and real-world datasets.",{"name":82,"@type":73,"acceptedAnswer":83},"Does the advantage of normed spaces depend on graph curvature or size?",{"text":84,"@type":76},"Empirical results confirm superiority for growing graph families associated with negative, zero, and positive curvature, and show robustness as graph sizes increase. Computational resource growth is also slower than for alternative Riemannian approaches.",{"name":86,"@type":73,"acceptedAnswer":87},"What applied tasks demonstrate the usefulness of normed space graph embeddings?",{"text":88,"@type":76},"The paper demonstrates utility on link prediction and recommender systems, with the l1 normed space surpassing baseline spaces in these applied graph embedding tasks.","https://schema.org",{"og:url":52,"og:type":91,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":93,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":106,"show_sort_weight":107,"slug":108},"Exam",70,"exam",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},5,"Comic",60,"comic",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},6,"Technology",50,"technology",{"id":120,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":122,"slug":123},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":125,"slug":126},30,"research-report",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":130,"slug":131},9,"Religion & Spirituality",20,"religion-spirituality",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":130,"slug":134},"World Cup","world-cup",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":136,"slug":138},10,"Lifestyle","lifestyle",{"id":140,"doc_module":4,"doc_module_name":46,"category_name":141,"show_sort_weight":110,"slug":142},19,"General","general"]