[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81827-en":3,"doc-seo-81827-105":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},81827,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Testing Unate Distributions","Study of unate distributions on {±1}^n is initiated as a natural analogue of unate Boolean functions, extending techniques from monotone distribution testing. Two core testing problems are addressed: uniformity testing for unate distributions and conditional-sampling-based testing of unateness within subcubes. The work provides efficient algorithms and associated lower bounds, including a weak learning subroutine for hidden coordinate orientations and a new correlation bound for the estimates. Results also show challenges beyond monotone reduction.","Testing Unate Distributions  \nDaeho Lee  \nMassachusetts Institute of Technology  \n[daeho0_0@mit. edu](daeho0_0@mit. edu)  \nShivam Nadimpalli Massachusetts Institute of Technology  \n[shivamn@mit. edu](shivamn@mit. edu)  \narXiv :2607 .0 1573v 1 [ cs .DS] 2 Jul 2026  \nMingda Qiao University of Massachusetts, Amherst  \n[mingda. qiao. cs@gmail. com](mingda. qiao. cs@gmail. com)  \nRonitt Rubinfeld Massachusetts Institute of Technology  \n[ronitt@csail. mit. edu](ronitt@csail. mit. edu)  \nJuly 3, 2026  \nAbstract  \nWe initiate the study of unate distributions over {±1}n—a natural analogue of unate Boolean functions—by considering two basic testing problems that parallel well-studied questions for monotone distributions:  \n• UficfoannierdifontmSrmanonoohlitdtoer,yneneSTecedODsistsastAirrnyib2,g oinut010iofc)Uonnsntara(RtestubtiDonfistheltredut(nd)iosSenarsmve:pdWleioeshomSTowpOltexChatity20;(tn3heA/d2aa) snamamlogaszpouekle,ssaprCzreobumsluemafj-,• UnatenconditioensalssTesting ofamples in theArbitrasubcubery Distconditiroibnalutmioonsdel: WOnetgive ahe othteersthearndt,heatveurysestes(rnt3/2hat)  \ndraws conditional samples in a similar fashion, namely from O(1)-dimensional subcubes, Outher mtamoluestgonstirothngithonavwmeeascsfaanreorse,c(ntt2lyhic/3sphhrw)coowoboulmpn temldleosinxbsici.fiΩ(nc(Inan)n2t(tC)lhysehasakoumamratpplebeearfcmrtoomodyrmpeelttlehx,atleithy,eSnacomTOiveOupCapr ulex20prniit25ofoyar)cm.ohif mofty toreenodsttueocrinnricgeliitteyoson a subroutine that “weakly” learns the hidden orientations of a unate distribution, together with a new correlation bound for these estimates. Both tools may be of independent interest in studying monotonicity and unateness over {±1}n.  \n1 Introduction  \nMany fundamental distribution testing tasks—uniformity testing being the canonical example—become infeasible in high dimensional settings such as the Boolean hypercube {±1}n, requiring exponentially many samples in the dimension [Can20, C+22] . A central theme in the field has therefore been to identify structured settings that admit efficient testers: examples include Bayesian networks [CDKS17, DP17], Markov random fields [DDK19, BBC+20], and structured truncations [DLNS24, DNS25] .  \nAmong such structural assumptions, one of the most extensively studied is monotonicity, the distributional analogue of the classical Boolean function property.1 In the Boolean function setting, monotonicity has played a central role in sublinear algorithms for nearly three decades: it was among the first properties studied in Boolean function testing and continues to be actively investigated to this day [BT96, GGL+00 , CS13 , CST14 , KMS18 , LRV22 , CDL+24 , BCS25 , LV25 , CCC+25] . In the distributional setting, monotonicity has similarly been studied in depth, leading to tight upper and lower bounds for tasks such as uniformity testing and property testing under various access models [BKR04, RS09 , ACS10 , BFRV11 , ADK15 , AGP+19 , RV20 , BLMT23 , CCR+25] .  \nA closely related property in the Boolean function setting is unateness. Informally, a Boolean function is unate if each coordinate is either always non-decreasing or always non-increasing. The problem of testing unateness of Boolean functions was introduced alongside monotonicity testing in [GGL+00], and has since received considerable attention [KS16, CS16 , CWX17a, CWX17b, CW19 , BCP+20] . Both monotonicity and unateness are fundamental structural properties that arise naturally across many domains, for example in social choice (where we can view a Boolean function f as a voting rule), economics, learning theory, and circuit design.  \nDespite extensive work on unate functions, the analogous notion for distributions has, to the best of our knowledge, not been studied. In this work, we initiate the study of unate distributions over {±1}n. The motivation for this is twofold:  \n• The study of monotone distributions has a well-developed literature precisely because monotonicity is a natural structural ","cbCaiiAVAGHiA8hY","https://ap.wps.com/l/cbCaiiAVAGHiA8hY","pdf",660379,6,1,32,"English","en",105,"# Introduction\n## Our Results\n## Problem Definitions and Key Challenges","[{\"question\":\"What is an unate distribution over {±1}^n?\",\"answer\":\"A distribution D over {±1}^n is unate if there exists an orientation vector σ such that the distribution of x ⊕ σ has a monotone probability mass function.\"},{\"question\":\"Which two main testing problems does the work focus on?\",\"answer\":\"It studies uniformity testing for unate distributions and testing unateness of an unknown distribution using conditional samples from subcubes.\"},{\"question\":\"Why can unate distribution testing not be reduced to the monotone case?\",\"answer\":\"The paper shows that even basic tasks like uniformity testing have different sample complexity and require new ideas beyond monotone reductions, reflecting the richer directional structure of unateness.\"}]","Testing Unate Distributions | PDF",1784176405,81,{"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},"testing-unate-distributions","",{"@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/testing-unate-distributions/81827/",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","2026-07-29","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 an unate distribution over {±1}^n?","Question",{"text":77,"@type":78},"A distribution D over {±1}^n is unate if there exists an orientation vector σ such that the distribution of x ⊕ σ has a monotone probability mass function.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"Which two main testing problems does the work focus on?",{"text":82,"@type":78},"It studies uniformity testing for unate distributions and testing unateness of an unknown distribution using conditional samples from subcubes.",{"name":84,"@type":75,"acceptedAnswer":85},"Why can unate distribution testing not be reduced to the monotone case?",{"text":86,"@type":78},"The paper shows that even basic tasks like uniformity testing have different sample complexity and require new ideas beyond monotone reductions, reflecting the richer directional structure of unateness.","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"]