[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-657489-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-657489-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","nearly-doubling-spaces-of-persistence-diagrams-abstract","Nearly-Doubling Spaces of Persistence Diagrams - Abstract","","The space of persistence diagrams under bottleneck distance is known to have infinite doubling dimension, which makes it hard to analyze the asymptotic performance of metric search algorithms on this space. The paper introduces nearly-doubling metrics, defined via Gromov-Hausdorff proximity to spaces with bounded doubling dimension, and proves bounded k-point persistence diagrams are nearly-doubling. This enables treating persistence diagrams like doubling metric spaces, including bounds for k-point bottleneck spaces across a general quotient-metric framework.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/nearly-doubling-spaces-of-persistence-diagrams-abstract/657489/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/nearly-doubling-spaces-of-persistence-diagrams-abstract/657489.png","ImageObject",300,407,{"name":42,"@type":43},"putri.kecil17","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":52,"interactionType":53,"userInteractionCount":33},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What problem does the paper address about persistence diagrams?","Question",{"text":62,"@type":63},"It addresses the fact that persistence diagrams under bottleneck distance have infinite doubling dimension, complicating analysis of metric search algorithms and data structures.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"What are nearly-doubling metrics?",{"text":67,"@type":63},"Nearly-doubling metrics are those that are Gromov-Hausdorff close to metric spaces with bounded doubling dimension.",{"name":69,"@type":60,"acceptedAnswer":70},"What main result is proved regarding persistence diagrams?",{"text":71,"@type":63},"The paper proves that bounded k-point persistence diagrams are nearly-doubling, allowing their behavior to resemble that of a doubling metric space for certain algorithmic analyses.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},657489,1793110236,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":92,"show_sort_weight":93,"slug":94},"Exam",70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},8,30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":111,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":33,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":145},962094509873,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Nearly-Doubling Spaces of Persistence Diagrams  \nDonald R. Sheehy \\# Ñ  \nDepartment of Computer Science, North Carolina State University, Raleigh, NC, USA  \nSiddharth S. Sheth  \nDepartment of Computer Science, North Carolina State University, Raleigh, NC, USA  \n~~ Abstract ~~  \nThe space of persistence diagrams under bottleneck distance is known to have infinite doubling dimension. Because many metric search algorithms and data structures have bounds that depend on the dimension of the search space, the high-dimensionality makes it difficult to analyze and compare asymptotic running times of metric search algorithms on this space.  \nWe introduce the notion of nearly-doubling metrics, those that are Gromov-Hausdorff close to metric spaces of bounded doubling dimension and prove that bounded k-point persistence diagrams are nearly-doubling. This allows us to prove that in some ways, persistence diagrams can be expected to behave like a doubling metric space. We prove our results in great generality, studying a large class of quotient metrics (of which the persistence plane is just one example) . We also prove boundson the dimension of the k-point bottleneck space over such metrics.  \nThe notion of being nearly-doubling in this Gromov-Hausdorff sense is likely of more general interest. Some algorithms that have a dependence on the dimension can be analyzed in terms of the dimension of the nearby metric rather than that of the metric itself. We give a specific example of this phenomenon by analyzing an algorithm to compute metric nets, a useful operation on persistence diagrams.  \n2012 ACM Subject Classification Theory of computation → Computational geometry  \nKeywords and phrases Topological Data Analysis, Persistence Diagrams, Gromov-Hausdorff Distance Digital Object Identifier 10.4230/LIPIcs.SoCG.2022.60  \nFunding This research was supported by the NSF under grant CCF-2017980 .  \n 1  Introduction  \nA persistence diagram is a topological summary commonly used in topological data analysis (TDA) . Ever since their introduction, persistence diagrams have been a popular tool to compare the shapes of point clouds, metric spaces, and real-valued functions.  \nA significant advantage of persistence diagrams over many other topological invariants is that they come equipped with a natural metric, the bottleneck distance, and thus topological features are rendered not only qualitative, but also quantitative. This opens the possibility of doing metric analysis on persistence diagrams, such as (approximate) nearest neighbor search or range search.  \nMany metric proximity search algorithms and data structures have asymptotic running time bounds in terms of the doubling dimension of the search space [6, 10] . The metric space of persistence diagrams with the bottleneck distance is known to have infinite doubling dimension [8], making it unclear whether one ought to apply standard data structures such as cover trees [1] or net trees [10] to search in this space. Although the space of all persistence diagrams is infinite-dimensional, all hope is not lost. In this paper, we show that the bottleneck space of bounded persistence diagram (i.e., those whose points are ina bounded region) is close in a Gromov-Hausdorff sense to a finite-dimensional space. Our approach is to consider a very general class of quotient metrics generalizing the persistence plane and then bound the doubling dimension of bottleneck distances over such metrics. We also show that for some algorithms whose running time depends on the doubling dimension,  \n© Donald R. Sheehy and Siddharth S. Sheth;  \nlicensed under Creative Commons License CC-BY 4.0  \n38th International Symposium on Computational Geometry (SoCG 2022) .  \nEditors: Xavier Goaoc and Michael Kerber; Article No. 60; pp. 60:1–60:15 Leibniz International Proceedings in Informatics  \n Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl  \nPublishing, Germany  \n60:2 Nearly-Doubling Spaces of Persistence Diagrams  ","cbCaincURT5ofGv2","https://ap.wps.com/l/cbCaincURT5ofGv2","pdf",823545,15,"English","# Introduction\n## Related Work","[{\"question\":\"What problem does the paper address about persistence diagrams?\",\"answer\":\"It addresses the fact that persistence diagrams under bottleneck distance have infinite doubling dimension, complicating analysis of metric search algorithms and data structures.\"},{\"question\":\"What are nearly-doubling metrics?\",\"answer\":\"Nearly-doubling metrics are those that are Gromov-Hausdorff close to metric spaces with bounded doubling dimension.\"},{\"question\":\"What main result is proved regarding persistence diagrams?\",\"answer\":\"The paper proves that bounded k-point persistence diagrams are nearly-doubling, allowing their behavior to resemble that of a doubling metric space for certain algorithmic analyses.\"}]","Nearly-Doubling Spaces of Persistence Diagrams - Abstract | PDF",1792981000,38]