[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81964-en":3,"doc-seo-81964-105":31,"detail-sidebar-cat-0-en-105":93},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81964,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Lower Bounds for Approximating the Vietoris-Rips Filtration","Lower bounds are established for approximating the Vietoris–Rips filtration VR(−), a central object in topological data analysis and persistent homology. Building on prior sparse approximation work under bounded doubling dimension, the results prove that the geometric assumption is necessary: using homotopy interleavings, exponential-size lower bounds arise for any fixed c in [1, √2), and superlinear lower bounds arise for any fixed c ≥ 1. The conclusions extend to the intrinsic Čech filtration and to any bifiltration containing VR(−) as a 1-parameter slice, including function-, degree-, and subdivision-Rips.","arXiv :2607 .06524v1 [math .AT] 7 Jul 2026  \nLOWER BOUNDS FOR APPROXIMATING THE VIETORIS-RIPS  \nFILTRATION  \nKENNETH MCCABE  \nAbstract. The Vietoris–Rips filtration VR (−) is a standard tool for analyzing the shape of data within topological data analysis. Beginning with seminal work of Sheehy, a substantial amount of research has centered on constructing linear-size sparse approximations to VR (−) and related filtrations for metric spaces of bounded doubling dimension. We show that this geometric assumption is necessary in a precise sense. Working in the framework of homotopy interleavings, we show that for any fixed c ∈ [1 , √2), there exists a family of finite metric spaces for which any finitely presented c-approximation to VR(−) has exponential size. We also show that for any fixed c ≥ 1, there exists a family of finite metric spaces for which any finitely presented c-approximation to VR (−) has superlinear size, yielding an obstruction to linear-size approximations for any fixed approximation factor. Both results extend to the intrinsic ˇCech filtration and to any bifiltration containing VR(−) as a 1-parameter slice, including the function-Rips, degree-Rips, and subdivision-Rips bifiltrations.  \n1. Introduction  \nFor a finite metric space X and scale r ≥ 0, the Vietoris-Rips complex VR (X)r is the simplicial complex whose simplices are the nonempty subsets ofX of diameter at most 2r. Letting r vary gives the Vietoris-Rips filtration VR(X) . The Vietoris-Rips filtration is a central tool within topological data analysis, typically via persistent homology computations on metric data, and has been studied extensively in its own right. However, as the k-skeleton of VR (X) has Θ(|X|k+1) simplices, direct computations can become infeasible as the size of X or the homological dimension of interest grows. This motivates the search for filtrations of asymptotically smaller size whose persistent homology closely approximates that of VR(−) .  \n1.1. Related Work. We use approximation informally throughout this section to mean a construction whose persistent homology is close to that of the target; the precise notion varies by paper. Our approximation results use the language of homotopy interleavings [8], a notion that implies approximation at the level of persistent homology, but is not implied by it. By an exact model of a filtration F we mean a functor valued in simplicial complexes that is weakly equivalent to F; see Section 2 for precise definitions.  \n1.1.1. Sparse Filtrations. For finite metric spaces X of bounded doubling dimension, Sheehy [38] showed that VR(X) admits a (1+ϵ)-approximation of size O(|X|) for any fixed ϵ > 0. Botnan and Spreemann [11] subsequently extended this result to ˇCech filtrations of Euclidean point clouds, and further improvements and variants have been developed in [13, 16, 18–20, 23, 39] . Complementing this line of work, Edelsbrunner et al. [25] showed via direct geometric arguments that for  \n2 KENNETH MCCABE  \npoint clouds in Rd , only a linear number of homological features in Vietoris-Ripsand ˇCech filtrations can persist over an interval of fixed length.  \nWithout geometric assumptions on X, the situation is more delicate. Choudhary et al. [19] achieved an O(polylog(|X|))-approximation to VR(X) of size |X| O(1) for arbitrary finite metric spaces X . The approximation factor, however, grows with |X| . Brun and Blaser [12] defined a (1 + ϵ)-approximation to ˇCech filtrations of point clouds in arbitrary metric spaces, extending the construction of [16], but did not give a formal size analysis. In a recent preprint, Leit˜ao [31] constructed a 3-approximation to VR (−) for arbitrary metric spaces, but also did not prove any size bounds.  \nAnalogous questions have been studied for multiparameter filtrations. The multicover bifiltration M(−), introduced by Sheehy [37] and studied further by Edelsbrunner and Osang [24] and Corbet et al. [21], is a density-sensitive [9] bifiltration for point sets X","cbCain6yOmzh85xX","https://ap.wps.com/l/cbCain6yOmzh85xX","pdf",384533,6,1,15,"English","en",105,"# Abstract\n# Introduction\n## Related Work\n### Sparse Filtrations\n### Lower Bounds","[{\"question\":\"What is the main problem addressed by the paper?\",\"answer\":\"The paper studies when the Vietoris–Rips filtration VR(−) can be approximated by smaller filtrations while still closely matching persistent homology.\"},{\"question\":\"Why do the authors consider bounded doubling dimension assumptions?\",\"answer\":\"Prior work shows linear-size sparse approximations under geometric assumptions such as bounded doubling dimension; the paper proves these assumptions are necessary in a precise sense via lower bounds.\"},{\"question\":\"What do the lower bound results imply for approximation size?\",\"answer\":\"For any fixed c in [1, √2), the paper constructs metric spaces where any finitely presented c-approximation must have exponential size; for any fixed c ≥ 1, there are spaces where any finitely presented c-approximation has superlinear size, blocking linear-size approximations at fixed accuracy.\"}]","Lower Bounds for Approximating the Vietoris-Rips Filtration | PDF",1784177310,38,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"lower-bounds-for-approximating-the-vietoris-rips-filtration","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/lower-bounds-for-approximating-the-vietoris-rips-filtration/81964/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-07-30","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What is the main problem addressed by the paper?","Question",{"text":77,"@type":78},"The paper studies when the Vietoris–Rips filtration VR(−) can be approximated by smaller filtrations while still closely matching persistent homology.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"Why do the authors consider bounded doubling dimension assumptions?",{"text":82,"@type":78},"Prior work shows linear-size sparse approximations under geometric assumptions such as bounded doubling dimension; the paper proves these assumptions are necessary in a precise sense via lower bounds.",{"name":84,"@type":75,"acceptedAnswer":85},"What do the lower bound results imply for approximation size?",{"text":86,"@type":78},"For any fixed c in [1, √2), the paper constructs metric spaces where any finitely presented c-approximation must have exponential size; for any fixed c ≥ 1, there are spaces where any finitely presented c-approximation has superlinear size, blocking linear-size approximations at fixed accuracy.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,112,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":108,"doc_module":4,"doc_module_name":47,"category_name":109,"show_sort_weight":110,"slug":111},5,"Comic",60,"comic",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":108,"slug":139},19,"General","general"]