[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-327139-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-327139-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","scaling-laws-for-grid-based-approximate-nearest-neighbor-search-in-high-dimensions","Scaling Laws for Grid-Based Approximate Nearest Neighbor Search in High Dimensions","","Grid-based approximate nearest neighbor (ANN) search lacks coverage in modern scaling analyses. The paper presents a systematic characterization of a multiprobe grid algorithm across dataset size N and dimensionality d. Experiments identify a previously unreported d-scaling crossover on the GloVe embedding family, where multiprobe grid preserves an approximately constant dimensional scaling exponent while other methods see throughput degradation. The approach achieves near-linear query scaling in N and lower indexing cost, suggesting competitiveness in rebuild-heavy or high-dimensional regimes where indexing expense and dimensional robustness dominate.",{"@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/scaling-laws-for-grid-based-approximate-nearest-neighbor-search-in-high-dimensions/327139/",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/scaling-laws-for-grid-based-approximate-nearest-neighbor-search-in-high-dimensions/327139.png","ImageObject",300,407,{"name":42,"@type":43},"Ava Thompson","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-09-21",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",7,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What scaling dimensions does the paper analyze for multiprobe grid search?","Question",{"text":63,"@type":64},"It analyzes scaling with respect to dataset size N and embedding dimensionality d for a multiprobe grid algorithm.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What is the reported d-scaling phenomenon in experiments?",{"text":68,"@type":64},"Experiments reveal a previously unreported d-scaling crossover on the GloVe embedding family, where multiprobe grid maintains an approximately constant dimensional scaling exponent.",{"name":70,"@type":61,"acceptedAnswer":71},"How does the paper argue multiprobe grid compares in query and indexing cost?",{"text":72,"@type":64},"It reports near-linear query scaling in N and lower indexing cost than competing ANN methods, motivating use in rebuild-heavy or high-dimensional settings.","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},327139,1790876346,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,110,114,119,122,126],{"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":55,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},8,30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":97,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":111,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":145,"preview_page_count":4},1649267921044,"https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1786009248482753345","arXiv :2607 .0 1283v 1 [ cs .LG] 1 Jul 2026  \nScaling Laws for Grid-Based Approximate Nearest Neighbor  \nSearch in High Dimensions  \nMatthew J. Liu  \nUniversity of California, Berkeley  \nWei Hang Zheng  \nUniversity of California, Berkeley  \nVidhan Purohit  \nUniversity of Toronto, St. George  \nSiqi Xie  \nIndependent Researcher  \nChieh-En Li  \nIndependent Researcher  \nJerry Li  \nUniversity of Waterloo  \nNoah Flynn  \nUniversity of California, Berkeley  \n[matthewliu@berkeley.edu](matthewliu@berkeley.edu)[ ](matthewliu@berkeley.edu)weihang [zheng@berkeley.edu](zheng@berkeley.edu)[ ](zheng@berkeley.edu)[viddyp1804@gmail.com](viddyp1804@gmail.com)[ ](viddyp1804@gmail.com)[siqixie@alumni.cmu.edu](siqixie@alumni.cmu.edu)[ ](siqixie@alumni.cmu.edu)[chiehenli@gmail.com](chiehenli@gmail.com)  \n[li.zhiyi.jerry@gmail.com](li.zhiyi.jerry@gmail.com)  \n[noahflynn@berkeley.edu](noahflynn@berkeley.edu)  \nAbstract  \nGrid-based approaches to approximate nearest neighbor (ANN) search have been absent from modern scaling analyses. We present a systematic characterization of a multiprobe grid algorithm with respect to dataset size N and dimensionality d. Our experiments reveal a previously unreported d-scaling crossover on the GloVe embedding family, in which multiprobe grid search maintains an approximately constant dimensional scaling exponent while other graph-, tree-, and partitioning-based methods exhibit degrading throughput.  \nThe advantage comes with near-linear query scaling in N, but also with lower indexing cost than competing ANN methods. Our results suggest that grid-based methods such as multiprobe grid may be competitive in rebuild-heavy or high-dimensional settings where indexing cost and dimensional robustness dictate performance. More broadly, recent work has formalized self-attention as an ANN operation. Thus, the N-and d-scaling properties of ANN algorithms may guide cost analysis of efficient transformer architectures. Code is available at: [https://github.com/weiz345/MultiProbeANN](https://github.com/weiz345/MultiProbeANN).  \n1. Introduction  \nApproximate nearest neighbor (ANN) search is central to modern machine learning systems operating on large, high-dimensional datasets Darrell et al. (2005) . Beyond its classical role in retrieval, ANN has emerged as a computational primitive in transformer architectures. Recent studies formalize self-attention as an ANN operation over token embeddings, enabling sub-quadratic approximations to full attention Haris (2025); Liu et al. (2025); Kang and Neto (2025) . As empirical scaling laws guide transformer design Kaplan et al. (2020), the scaling behavior of ANN algorithms is increasingly relevant to the development of more efficient transformer architectures. In particular, scaling dataset size N stresses candidate  \n© M.J. Liu, W.H. Zheng, V. Purohit, S. Xie, C.-E. Li, J. Li & N. Flynn.  \nLiu Zheng Purohit Xie Li Li Flynn  \nfiltering and indexing efficiency, while scaling dimensionality d exacerbates the curse of dimensionality and degrades the effectiveness of geometric pruning. As a result, ANN methods including graph-, tree-, and partitioning-based approaches exhibit regime-dependent trade-offs in which no method dominates universally Xiao et al. (2024); Iff et al. (2026) .  \nGrid-based methods (a variant of partitioning approaches) played a foundational role in early ANN theory Indyk and Motwani (1998); Chan (1997), but remain undercharacterized relative to graph-and tree-based methods in modern scaling analyses (see Appendix A for Related Work) . In this work, we characterize N and d scaling relationships for a multiprobe grid algorithm (overview in Section 2) that decouples cell selection from d, where cellselection denotes the query-time procedure that identifies which grid cells supply candidate vectors. This decoupling is achieved by performing cell selection in a PCA-reduced subspace Rm while re-ranking candidates in Rd. We show that while multiprobe grid exhibits nearlinear sc","cbCaivrG0pDM3wQr","https://ap.wps.com/l/cbCaivrG0pDM3wQr","pdf",1075252,34,"English","# 1. Introduction\n## ANN as a primitive in transformer architectures\n## Grid-based methods and the multiprobe grid approach\n# 2. Theoretical scaling model for multiprobe grid search\n## Cost model\n## Recall model\n## Closed-form relationship\n## Log-linearity\n# 3. Empirical scaling laws","[{\"question\":\"What scaling dimensions does the paper analyze for multiprobe grid search?\",\"answer\":\"It analyzes scaling with respect to dataset size N and embedding dimensionality d for a multiprobe grid algorithm.\"},{\"question\":\"What is the reported d-scaling phenomenon in experiments?\",\"answer\":\"Experiments reveal a previously unreported d-scaling crossover on the GloVe embedding family, where multiprobe grid maintains an approximately constant dimensional scaling exponent.\"},{\"question\":\"How does the paper argue multiprobe grid compares in query and indexing cost?\",\"answer\":\"It reports near-linear query scaling in N and lower indexing cost than competing ANN methods, motivating use in rebuild-heavy or high-dimensional settings.\"}]","Scaling Laws for Grid-Based Approximate Nearest Neighbor Search in High Dimensions | PDF",1789976518,86]