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The talk then develops topologically optimal and information-theoretically optimal mixtures using sweep-based constructions and local perturbations, followed by topological density estimation (TDE) for persistent bandwidth selection and (de/re)blurring using a Gaussian kernel semigroup to maintain topological optimality.",{"@graph":14,"@context":73},[15,34,56],{"@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/topological-mixture-estimation-2018-symposium-on-data-science-and-statistics/201461/",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/topological-mixture-estimation-2018-symposium-on-data-science-and-statistics/201461.png","ImageObject",300,407,{"name":42,"@type":43},"Dipper","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-27","2026-09-04",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",12,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What does “topological mixture estimation” aim to compute?","Question",{"text":63,"@type":64},"It aims to produce mixture models (and related density estimates) whose components preserve a topological predicate while maintaining desirable optimality criteria across scales or objectives.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How do undersmoothing and oversmoothing affect the computed persistent structure?",{"text":68,"@type":64},"Undersmoothing yields a situation where the initial persistent counts match the sample size, while oversmoothing can collapse structure to a single persistent value; despite this, persistent structure is described as stable under perturbations.",{"name":70,"@type":61,"acceptedAnswer":71},"How is bandwidth selected in the topological density estimation approach?",{"text":72,"@type":64},"TDE computes a topological invariant from kernel density estimates at various scales, selects the most persistent value, and sets the bandwidth using a median rule based on that persistence summary.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},201461,1788534083,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":25,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":112,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":113,"slug":114},8,30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":97,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":112,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":140,"language":141,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":12,"update_tm":80,"read_time":145,"preview_page_count":4},1374404997633,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Topological mixture estimation  \nSteve Huntsman  \nFAST Labs ([goo.gl/zWw5HQ](goo.gl/zWw5HQ)) Cyber Technology  \n17 May 2018  \nApproved for public release; distribution unlimited  \n2018 Symposium on Data Science and Statistics Topological mixture estimation 2  \n The rudiments of topological data analysis  \n􀀏 Approximate X = (Xj)nj=1 􀀚 (Rd )n at various scales h > 0  \n􀀏 Compute a topological invariant of the approximations  \n􀀏 Highlight invariants that persist as h varies  \n􀀏 Toy example: 12 equispaced points on S 1 (􀀁 􀀙 0:2588)  \n􀀏 􀀌0 = 12 􀀁 1[0 ;􀀁) + 1 [􀀁 ; 1) + 1 [1 ;1)  \n􀀏 􀀌1 = 1 [􀀁 ; 1)  \nh = 0:2   \nh = 0:3  \nApproved for public release; distribution unlimited  \n2018 Symposium on Data Science and Statistics Topological mixture estimation 3  \n Belaboring some details  \n􀀏 There exists \" > 0 s.t. 􀀌0 ([0; \")) = n (undersmoothing)  \n􀀏 􀀌0 ([diam X ; 1)) = 1 (oversmoothing)  \n􀀏 Persistent structure is stable under perturbations  \n􀀏 Example: n = 100, 􀀎 = 0:05, j Xj 􀀘 U([1 􀀀 􀀎; 1 + 􀀎]), \"0 􀀙 􀀎  \n􀀏 􀀌0 = n 􀀁 1[0;\") + noise0 􀀁 1[\";􀀁) + 1 [􀀁 ; 1􀀀\"0 ) + 1 [1􀀀\"0 ;1)  \n􀀏 􀀌1 = noise1 􀀁 1[\";􀀁) + 1 [􀀁 ; 1􀀀\"0 )  \nApproved for public release; distribution unlimited  \nIs topological data analysis interesting or useful for d = 1?  \nYes and YES  \nApproved for public release; distribution unlimited  \n2018 Symposium on Data Science and Statistics Topological mixture estimation 5  \n Approximate: kernel density estimate  \n􀀏 X 􀀘 f 􀀙 ^fh;X := ~~1~~n Pnj=1 KXj ;h with K􀀖;h(x) := ~~1~~h K 􀀀 ~~x ~~~~􀀖~~ 􀀁  \n􀀏 To estimate f , just select a good bandwidth h  \n􀀏 Choice of K much less important than h as long as K is nice  \n􀀏 Least-squares cross-validation (CV) optimizes L2 error estimate without a priori information about f  \n􀀏 Epanechnikov is L2 optimal choice for K , but see above  \n􀀏 Gaussian is also good and very convenient  \n􀀏 If f is multimodal but we have no a priori information, the literature doesn't o􀀋er any signi􀀌cant improvement on CV  \nApproved for public release; distribution unlimited  \n2018 Symposium on Data Science and Statistics Topological mixture estimation 6  \n Compute: topologically optimal mixture  \n􀀏 A PDF f 2 C (Rd ) is unimodal i􀀋 f 􀀀1([y ; 1)) is contractible (i.e. , homotopic to a point) for all 0 \u003C y 􀀔 max f  \n􀀏 For d = 1, contractible = interval (captures intuitive notion)  \n􀀏 ucat(f ) is the smallest number M s.t. f = PMm=1 􀀙 mpm for some 􀀙 > 0, Pm 􀀙 m = 1, and pm unimodal: write (􀀙; p) j= f  \n􀀏 Lemma: ucat is invariant under homeomorphism  \n􀀏 Theorem [Baryshnikov and Ghrist]: For d = 1, a \\sweep\"algorithm e􀀎ciently produces a (􀀙; p) j= f  \nApproved for public release; distribution unlimited   \n2018 Symposium on Data Science and Statistics Topological mixture estimation 7  \n Compute: information theoretically optimal mixture  \n􀀏 Mixture (􀀙; p) j= f produced by sweep algorithm is ugly  \n􀀏 Want to preserve predicate (􀀙; p) j= f while maximizing mutual info J (􀀙; p) between 􀀄 􀀘 􀀙 and X 􀀘 Pm 􀀙 mpm  \n􀀏 Two lemmas regarding local perturbations of piecewise a􀀎ne (􀀙; p) that preserve (􀀙; p) j= f:  \n􀀏 J is convex  \n􀀏 Explicit tight bounds on perturbations  \n􀀏 Theorem: Greedy unimodality-preserving local perturbations obtain arg max(􀀙;p)j= f J (􀀙; p) in O(ucat 􀀁 jmeshj) iterations  \nApproved for public release; distribution unlimited   \n2018 Symposium on Data Science and Statistics Topological mixture estimation 8  \n Compute: information theoretically optimal mixture  \n􀀏 Mixture (􀀙; p) j= f produced by sweep algorithm is ugly  \n􀀏 Want to preserve predicate (􀀙; p) j= f while maximizing mutual info J (􀀙; p) between 􀀄 􀀘 􀀙 and X 􀀘 Pm 􀀙 mpm  \n􀀏 Two lemmas regarding local perturbations of piecewise a􀀎ne (􀀙; p) that preserve (􀀙; p) j= f:  \n􀀏 J is convex  \n􀀏 Explicit tight bounds on perturbations  \n􀀏 Theorem: Greedy unimodality-preserving local perturbations obtain arg max(􀀙;p)j= f J (􀀙; p) in O(ucat 􀀁 jmeshj) iterations  \nApproved for public release; distribution unlimited   \n2018 Symposium on Data Science and Statistics Topological mixture esti","cbCaimjVBxegzaqC","https://ap.wps.com/l/cbCaimjVBxegzaqC","pdf",1595014,18,"English","# The rudiments of topological data analysis\n## Approximate invariants across scales\n# Belaboring some details\n## Undersmoothing vs oversmoothing\n## Stability of persistent structure\n# Is topological data analysis interesting or useful for d = 1?\n# Approximate: kernel density estimate\n## Bandwidth choice via least-squares cross-validation\n# Compute: topologically optimal mixture\n## Unimodality and contractibility for d = 1\n# Compute: information theoretically optimal mixture\n## Convexity and greedy local perturbations\n# Topological density estimation (TDE)\n## Persistent value selection and bandwidth estimate\n# (De/re)blurring\n## Using kernel semigroup and preserving optimality","[{\"question\":\"What does “topological mixture estimation” aim to compute?\",\"answer\":\"It aims to produce mixture models (and related density estimates) whose components preserve a topological predicate while maintaining desirable optimality criteria across scales or objectives.\"},{\"question\":\"How do undersmoothing and oversmoothing affect the computed persistent structure?\",\"answer\":\"Undersmoothing yields a situation where the initial persistent counts match the sample size, while oversmoothing can collapse structure to a single persistent value; despite this, persistent structure is described as stable under perturbations.\"},{\"question\":\"How is bandwidth selected in the topological density estimation approach?\",\"answer\":\"TDE computes a topological invariant from kernel density estimates at various scales, selects the most persistent value, and sets the bandwidth using a median rule based on that persistence summary.\"}]","Topological mixture estimation - 2018 Symposium on Data Science and Statistics | PDF",45]