[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-665336-105":59,"doc-detail-665336-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","higher-order-persistence-diagrams","Higher-Order Persistence Diagrams","","Persistent homology tracks the appearance and disappearance of topological features across filtered data, encoding each feature’s lifetime as a persistence interval. In multiple instance learning, when all instances in a bag are filtered over the same ordered parameter space, interval relationships can be compared across instances. No existing approach aggregates persistence diagrams in MIL into a single diagram capturing implications between intervals. This work introduces higher-order persistence diagrams that recursively build higher-order intervals from pairs of lower-order intervals and then aggregate these relationships within each bag. Because explicit aggregation is computationally costly, harmonic aggregation computes aggregate coordinates via Fourier analysis without constructing the aggregate. Experiments on random network models and graph-classification benchmarks show harmonic aggregation attains the highest accuracy on the random model task and strong results on most real datasets.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/higher-order-persistence-diagrams/665336/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/higher-order-persistence-diagrams/665336.png","ImageObject",300,407,{"name":92,"@type":93},"awa","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What problem does the paper address in multiple instance learning with persistence diagrams?","Question",{"text":112,"@type":113},"It addresses the lack of an existing method that aggregates persistence diagrams in MIL into a single diagram representing implications between persistence intervals without converting them into vector or scalar forms.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are higher-order persistence diagrams constructed?",{"text":117,"@type":113},"Each higher-order persistence interval is built from a pair of lower-order persistence intervals, and the same construction is applied recursively to define diagrams of increasing order.",{"name":119,"@type":110,"acceptedAnswer":120},"Why introduce harmonic aggregation, and what does it improve?",{"text":121,"@type":113},"Explicit aggregation is computationally expensive due to quadratic pairing at each recursive step. Harmonic aggregation computes coordinates in frequency space directly from lower-order inputs using Fourier analysis, reducing asymptotic computational complexity.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},665336,1793298358,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},3985747858093,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","arXiv : 2605 .09866v2 [cs .CG] 25 Oct 2026  \nHigher-Order Persistence Diagrams  \nCharles Fanning  and Mehmet Aktas   \nSchool of Data Science and Analytics, Kennesaw State University, 1000 Chastain Rd NW, Kennesaw, 30144, Georgia, United States.  \n*Corresponding author(s). E-mail(s): [cfannin8@students.kennesaw.edu](cfannin8@students.kennesaw.edu) ;  \nContributing [authors:](authors: maktas1@kennesaw.edu)[ maktas1@kennesaw.edu](authors: maktas1@kennesaw.edu);  \nAbstract  \nPersistent homology tracks when topological features appear and disappear across filtered data and represents the lifetime of each feature by a persistence interval. In multiple instance learning, when the instances in a bag use filtrations over the same ordered parameter space, we can compare relationships between persistence intervals across instances. No existing method is designed to aggregate collections of persistence diagrams in multiple instance learning into a single persistence diagram of implications between intervals. We introduce higher-order persistence diagrams to represent relationships between persistence intervals and recursively construct each higher-order persistence interval from a pair of lowerorder persistence intervals. We then aggregate these relationships across the persistence diagrams in a bag. Explicit construction of this aggregate is computationally expensive, so we introduce harmonic aggregation to compute coordinates of the aggregate from the lower-order persistence diagrams without explicitly constructing it. Computing a harmonic aggregation coordinate has lower asymptotic computational complexity than explicitly constructing the aggregate. We evaluate harmonic aggregation against persistence landscapes, persistence images, and PersLay on random network model and real-world graph-classification benchmarks, and find that harmonic aggregation has the highest accuracy on the random network model benchmark and on most real-world data benchmarks.  \nSource code is available at [https://github.com/cfanning8/higher-order-PDs](https://github.com/cfanning8/higher-order-PDs).  \n1 Introduction  \nIn multiple instance learning (MIL), a label is assigned to a collection of observations rather than to each observation individually; this collection is called a bag, whose  \n1  \nobservations are instances [1–3] . While instances in MIL may generally take many different forms, we are interested in the case in which each instance is a persistence diagram. To represent each instance, we use persistent homology to compute a persistence diagram that records the birth and death parameters of topological features [4–6] . In this setting, each bag contains one persistence diagram for each instance. When we filter every instance in a bag over the same space of parameters, the persistence intervals lie in one common space, so the same relationship between persistence intervals has the same interpretation across instances.  \nMany MIL models aggregate vector-valued representations of instances [2, 3], and persistence-diagram vectorizations provide a direct way to apply these models to instances represented as persistence diagrams. Persistence landscapes (PL), persistence images (PI), and PersLay are standard persistence-diagram vectorizations [7–10] . Existing MIL methods that use persistent homology vectorize or scalarize the persistence diagrams before or during aggregation [11–16] . To our knowledge, no existing MIL method directly aggregates persistence diagrams without ever converting the diagrams into vector-or scalar-valued representations.  \nIn this paper we aggregate persistence diagrams into a higher-order persistence diagram without converting them into vector- or scalar-valued representations. We introduce higher-order persistence diagrams to represent relationships between lowerorder persistence intervals. We construct each higher-order persistence interval from a pair of lower-order persistence intervals and apply the same construction recursi","cbCaitoLc8RIU2CN","https://ap.wps.com/l/cbCaitoLc8RIU2CN","pdf",2905589,23,"English","# Abstract\n# 1 Introduction\n# 2 Preliminaries","[{\"question\":\"What problem does the paper address in multiple instance learning with persistence diagrams?\",\"answer\":\"It addresses the lack of an existing method that aggregates persistence diagrams in MIL into a single diagram representing implications between persistence intervals without converting them into vector or scalar forms.\"},{\"question\":\"How are higher-order persistence diagrams constructed?\",\"answer\":\"Each higher-order persistence interval is built from a pair of lower-order persistence intervals, and the same construction is applied recursively to define diagrams of increasing order.\"},{\"question\":\"Why introduce harmonic aggregation, and what does it improve?\",\"answer\":\"Explicit aggregation is computationally expensive due to quadratic pairing at each recursive step. Harmonic aggregation computes coordinates in frequency space directly from lower-order inputs using Fourier analysis, reducing asymptotic computational complexity.\"}]","Higher-Order Persistence Diagrams | PDF",58]