[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86023-en":3,"doc-seo-86023-105":30,"detail-sidebar-cat-0-en-105":91},{"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":13,"seo_description":14,"update_tm":28,"read_time":29},86023,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","The VC Dimension of Partial Concept Classes via Radon’s Theorem","The paper defines partial concept classes (PCCs) as families of partial functions mapping a ground set to {0,1,*}, where * forms a grey region. VC dimension is studied through sets that can be shattered without ever using the * value. Two geometric PCCs in real Banach spaces are analyzed under a margin ε>0: expanded half-spaces (grey strip next to a half-space) and expanded balls (grey annulus around a unit ball). Dimension-free upper and matching lower bounds are proved, extending prior Euclidean results, and a Dense Neighborhood Lemma is derived in Lp spaces using a nodimensional Radon-based method.","arXiv :2607 . 1075 1v 1 [ cs .LG] 12 Jul 2026  \nTHE VC DIMENSION OF PARTIAL CONCEPT CLASSES VIA RADON’S  \nTHEOREM  \nGRIGORY IVANOV, ATTILA JUNG, AND M´ARTON NASZ´ODI  \nAbstract. Following Alon, Hanneke, Holzman, and Moran (FOCS 2021), we define a partial concept class (PCC) as a family of partial functions 􀁦 : 􀁖 → {0, 1 , *}; equivalently, its concepts partition the ground set into black (􀁦−1(1)), grey (􀁦−1(*)), and white parts (􀁦−1(0)) . Its VC dimension is defined by shattering sets on which the value * is not taken. We study two geometric PCCs in real Banach spaces, both with a margin 􀀎 > 0: expanded half-spaces, where the grey part is a strip of width at least 􀀎 adjacent to a half-space, and expanded balls, where the grey part is an annulus of width 􀀎 around a unit radius ball.  \nOur main results are dimension-free upper bounds on the VC dimension of the PCC of expanded balls in 􀁌􀁰 (􀀖), 1 ≤ 􀁰 \u003C ∞ , including the non-Euclidean and algorithmically particularly relevant case ℓ􀁤1 . These bounds depend on the margin and on the radii, but not on the ambient dimension or the underlying measure space. These are extensions of the work of Bourneuf, Charbit, and Thomass´e (FOCS 2025) who studied the PCC of expanded balls in Euclidean space, that is, ℓ􀁤2 . We also prove lower bounds on the VC dimension that match the upper bounds in terms of the margin parameter 􀀎 . Finally, we derive a Dense Neighborhood Lemma in 􀁌􀁰 -spaces, again extending the known Euclidean results.  \nOur method relies on the linearization of the distance through a map into a space of non-trivial Rademacher type, and then the use of a balanced signed-sum estimate, or a nodimensional Radon theorem. The arguments rely on ideas from functional analysis that are clearly explained for the non-expert in that field.  \nIn studying 􀁌􀁰 (􀀖) spaces, for 1 ≤ 􀁰 ≤ 2 we use Schoenberg’s embedding theorem for metrics of negative type. For 􀁰 > 2, we introduce an entirely novel tool which we call the Taylor– Schoenberg lift which linearizes the 􀁰-th power of the distance.  \n1. Introduction  \nPartial concept classes and their VC dimension. The Vapnik–Chervonenkis (VC) dimension [1] is among the most fundamental complexity measures of a set system. Recall that a set system (or concept class) on a ground set 􀁖 is a family 􀁃 ⊆ 2􀁖 ; a finite set 􀁙 ⊆ 􀁖 is shattered by 􀁃 if every subset of 􀁙 is cut out by a member of 􀁃, i.e. {􀁃 ∩ 􀁙 : 􀁃 ∈ 􀁃} = 2􀁙 , and the VC dimension of 􀁃 is the largest cardinality of a shattered set. Its finiteness implies uniform convergence of random samples and PAC learnability (probably approximately correct learning), and it is a workhorse of combinatorics and geometry through 􀀢-nets, the (􀁰,􀁱)-theorem, and range-searching, among many others.  \nAlon, Hanneke, Holzman and Moran [8] recently introduced partial concept classes. A partial concept on 􀁖 is a function 􀁣 : 􀁖 → {0 , 1 , *}; the value * marks the points on which 􀁣 is left  \nDate: July 7, 2026 .  \n2020 Mathematics Subject Classification. 68Q32, 52A35, 46B07, 68R05, 05D40 .  \nKey words and phrases. VC dimension; partial concept class; tri-graph; tri-hypergraph; Rademacher type; no-dimensional Radon and Carath´eodory theorems; fat-shattering (scale-sensitive) dimension; margin; expanded half-spaces and balls; Dense Neighborhood Lemma; 􀀢-net; negative-type embedding.  \n[G.I. is](G.I. is) supported by Projeto Paz and Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior-Brasil (CAPES) -23038.015548/2016-06 .  \n[A.J. is](A.J. is) supported by the ERC Advanced Grant no. 101054936 “ERMiD” as well as NRDI EXCELLENCE-24 grant no. 151504 Combinatorics and Geometry.  \n[M.N. is](M.N. is) supported by National Research Development and Innovation Fund grant 143778, the ELTE TKP 2021-NKTA-62 funding scheme, grant 2024-1 .2.8-T´ET-IPARI-CN-2025-00011 as well as NRDI EXCELLENCE- 24 grant no. 151504 Combinatorics and Geometry.  \n2 GRIGORY IVANOV, ATTILA JUNG, AND M´ARTON NASZ´ODI  \nundefined. Writing 􀁣 as the ordered trip","cbCaiiNldx33PRUe","https://ap.wps.com/l/cbCaiiNldx33PRUe","pdf",431644,4,1,22,"English","en",105,"# Introduction\n## Partial concept classes and VC dimension\n## Expanded half-spaces and expanded balls\n# Main results and method","[{\"question\":\"What is a partial concept class (PCC) and how is its VC dimension defined?\",\"answer\":\"A PCC is a family of partial functions χ:X→{0,1,*}. Concepts induce a partition into black (χ^{-1}(1)), grey (χ^{-1}(*)), and white (χ^{-1}(0)); the VC dimension is defined by shattering sets where the value * is not taken.\"},{\"question\":\"What geometric PCCs are studied in the paper?\",\"answer\":\"The paper studies two geometric PCCs in real Banach spaces with margin ε\\u003e0: expanded half-spaces, where the grey region is a strip adjacent to a half-space, and expanded balls, where the grey region is an annulus around a unit radius ball.\"},{\"question\":\"What do the main results say about VC dimension?\",\"answer\":\"The work provides dimension-free upper bounds on the VC dimension for expanded balls in L_p spaces (1≤p\\u003c∞), depending on the margin ε and radii but not on ambient dimension or the underlying measure space. It also proves lower bounds that match the upper bounds in their dependence on ε, and derives a Dense Neighborhood Lemma in L_p-spaces.\"}]",1784207871,55,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"the-vc-dimension-of-partial-concept-classes-via-radons-theorem","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"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":20},"https://docshare.wps.com/document/the-vc-dimension-of-partial-concept-classes-via-radons-theorem/86023/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is a partial concept class (PCC) and how is its VC dimension defined?","Question",{"text":75,"@type":76},"A PCC is a family of partial functions χ:X→{0,1,*}. Concepts induce a partition into black (χ^{-1}(1)), grey (χ^{-1}(*)), and white (χ^{-1}(0)); the VC dimension is defined by shattering sets where the value * is not taken.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What geometric PCCs are studied in the paper?",{"text":80,"@type":76},"The paper studies two geometric PCCs in real Banach spaces with margin ε>0: expanded half-spaces, where the grey region is a strip adjacent to a half-space, and expanded balls, where the grey region is an annulus around a unit radius ball.",{"name":82,"@type":73,"acceptedAnswer":83},"What do the main results say about VC dimension?",{"text":84,"@type":76},"The work provides dimension-free upper bounds on the VC dimension for expanded balls in L_p spaces (1≤p\u003C∞), depending on the margin ε and radii but not on ambient dimension or the underlying measure space. It also proves lower bounds that match the upper bounds in their dependence on ε, and derives a Dense Neighborhood Lemma in L_p-spaces.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]