[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85756-en":3,"doc-seo-85756-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85756,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Learning Partition Trees for Nearest Neighbor Search","Nearest neighbor search is studied through data-driven algorithm design, aiming to learn an efficient data structure for queries drawn from a known query distribution. The focus is on balanced halfspace trees that generalize space-partitioning methods such as locality-sensitive hashing. Under Gaussian-like marginal conditions and assuming a perfect tree exists, the paper gives an algorithm with nearly (|N|^ε) query time. Its core is the balanced halfspace cut problem: optimal proper learning is NP-hard without distributional assumptions, so an efficient improper learning method is developed using polynomial threshold representations.","arXiv :2607 .09909v 1 [ cs .DS] 10 Jul 2026  \nLearning Partition Trees for Nearest Neighbor Search  \nSanjeev Khanna∗ Ashwin Padaki† Erik Waingarten‡  \nJuly 14, 2026  \nAbstract  \nWe study nearest neighbor search from the perspective of data-driven algorithm design: given adataset 􀀥 ⊂ ℝ􀀳 of size 􀀽 and sample access to a query distribution over ℝ􀀳, the goal is to learn a data structure optimized for queries drawn from that specific distribution. We focus on the class of balanced halfspace trees, which naturally abstracts space-partitioning frameworks like locality-sensitive hashing. Assuming Gaussian-like marginal conditions on the dataset and query distribution, we give an efficient algorithm that learns a tree achieving 􀀾(􀀽􀀳) query time, provided that a perfect tree exists.  \nAt the core of our algorithmic approach is the balanced halfspace cut problem, where we are given a distribution over ℝ􀀳 × ℝ􀀳 and must find a balanced halfspace that minimizes the fraction of cut pairs. We prove that without distributional assumptions, finding the optimal balanced halfspace is NP-hard. To circumvent this computational barrier, we design an efficient improper learning algorithm: if the  \noptimalfunctionhoaflfsdpegarceeet(s1n2)􀁕tfrahatction of pacuts at moirsts, oanlg~~􀁕~~oth~~􀁙~~)mfroutactipuonts a balanced polynomial threshold  \n∗ NYU. Supported in part by National Science Foundation (NSF) award CCF-2625203 and AFOSR award FA9550-25-1-0107 .{[sanjeev.khanna@nyu.edu](sanjeev.khanna@nyu.edu})[}](sanjeev.khanna@nyu.edu})  \n†University of Pennsylvania. Supported by the National Science Foundation (NSF) GRFP under Grant No. DGE-2236662, and [Grant No. CCF-2337993.](Grant No. CCF-2337993. {apadaki@seas.upenn.edu})[ {](Grant No. CCF-2337993. {apadaki@seas.upenn.edu})[apadaki@seas.upenn.edu](Grant No. CCF-2337993. {apadaki@seas.upenn.edu})[}](Grant No. CCF-2337993. {apadaki@seas.upenn.edu})  \n‡University of Pennsylvania. Supported by the National Science Foundation (NSF) under Grant No. CCF-2337993 .{[ewaingar@seas.upenn.edu](ewaingar@seas.upenn.edu})[}](ewaingar@seas.upenn.edu})  \nContents  \n1 Introduction 3  \n2 Technical Overview 7  \n3 Preliminaries 10  \n4 Hardness of Proper Learning: Theorem 2 10  \n5 An Improper Learning Algorithm: Theorem 3 14  \n6 Learning Balanced Halfspace Cut Trees 21  \nA Additional Preliminaries 32  \nB Approximating Geometric Regions with Polynomials 32  \nC Omitted Proofs from Section 5 41  \nD Omitted Proofs from Section 6 46  \n1 Introduction  \nThe focus of this paper is nearest neighbor search. For a dimensionality 􀀳 ∈ ℕ, we receive as input a dataset of 􀀽 points 􀀥 = {􀀿1, . . . , 􀀿 􀀽} ⊂ ℝ􀀳 . We aim to build a data structure which can support nearest neighbor queries: given a query point 􀁀 ∈ ℝ􀀳, return the point 􀀿 ∈ 􀀥 minimizing ∥􀁀 − 􀀿 ∥2 over all points in the dataset.1 The goal is to design data structures that answer queries in time that is sublinear in 􀀽 (i.e., significantly faster than scanning the entire dataset) while keeping the space complexity manageable, ideally near-linear in 􀀽 .  \nNearest neighbor search has been studied extensively from the theoretical computer science perspective, dating back to the work of Minsky and Papert [MP69] and Knuth [Knu73], continuing throughout the 80sand 90s [LT80, Sam84, Cla88, Mei93], and leading to relatively recent developments [IM98, AI06, AR15, ALRW17, Rub18] . Historically, the driving question has been to understand the conditions under which one can achieve sublinear-time queries using polynomial space for worst-case datasets and queries. This line of work has yielded powerful algorithmic frameworks and techniques in settings of (i) low dimensionality,(ii) low intrinsic dimensionality, and (iii) approximate nearest neighbors in high dimensions (see [AIR18] for a more thorough overview) .  \nIn this work, we consider beyond worst-case data structures for nearest neighbor search, specifically taking the perspective of data-driven algorithm design [GR17, Bal21] . Our motivat","cbCailTgOgj7Mq2C","https://ap.wps.com/l/cbCailTgOgj7Mq2C","pdf",814412,1,47,"English","en",105,"# Introduction\n# Technical Overview\n# Preliminaries\n# Hardness of Proper Learning\n# An Improper Learning Algorithm\n# Learning Balanced Halfspace Cut Trees","[{\"question\":\"What problem does the paper address for nearest neighbor search?\",\"answer\":\"It considers learning a nearest neighbor data structure from sample access to a dataset and a query distribution, optimizing performance for queries drawn from that same distribution.\"},{\"question\":\"Why are balanced halfspace trees central to the approach?\",\"answer\":\"Balanced halfspace trees model space-partitioning frameworks, and the paper designs learning algorithms specifically for this class to achieve efficient query time.\"},{\"question\":\"What computational hardness result is proved, and how is it handled?\",\"answer\":\"Without distributional assumptions, finding the optimal balanced halfspace is NP-hard. The paper avoids this by designing an efficient improper learning algorithm using polynomial threshold representations.\"}]",1784206037,118,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"learning-partition-trees-for-nearest-neighbor-search","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/learning-partition-trees-for-nearest-neighbor-search/85756/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 problem does the paper address for nearest neighbor search?","Question",{"text":75,"@type":76},"It considers learning a nearest neighbor data structure from sample access to a dataset and a query distribution, optimizing performance for queries drawn from that same distribution.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why are balanced halfspace trees central to the approach?",{"text":80,"@type":76},"Balanced halfspace trees model space-partitioning frameworks, and the paper designs learning algorithms specifically for this class to achieve efficient query time.",{"name":82,"@type":73,"acceptedAnswer":83},"What computational hardness result is proved, and how is it handled?",{"text":84,"@type":76},"Without distributional assumptions, finding the optimal balanced halfspace is NP-hard. 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