[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84010-en":3,"doc-seo-84010-105":30,"detail-sidebar-cat-0-en-105":92},{"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},84010,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Quantifying and Expanding the Theoretical Capacity of Late-Interaction Retrieval Models","Late-interaction retrieval models using MaxSim similarity have shown strong empirical results, yet their theoretical representational power and relationship to other retrieval similarities remain unclear. This work constructs an exact replication of the inner product between any two non-negative k-sparse vectors, potentially with infinite dimension, using only O(k) representation space. It further identifies similarities expressible by MaxSim that fixed-size inner products cannot capture. Building on this, Signed MaxSim exactly replicates any real-valued inner product, while standard MaxSim cannot. Logical and aggregation interpretations connect MaxSim to soft-OR and positive CNF evaluation, and experiments with negation-rich queries validate gains in out-of-domain performance.","Quantifying and Expanding the Theoretical Capacity of Late-Interaction Retrieval Models  \nJulian Killingback  \nCenter for Intelligent Information Retrieval University of Massachusetts Amherst [jkillingback@cs.umass.edu](jkillingback@cs.umass.edu)  \nVarad Ingale  \nCenter for Intelligent Information Retrieval University of Massachusetts Amherst [vingale@umass.edu](vingale@umass.edu)  \nHamed Zamani  \nCenter for Intelligent Information Retrieval University of Massachusetts Amherst [zamani@cs.umass.edu](zamani@cs.umass.edu)  \nCameron Musco  \nUniversity of Massachusetts Amherst [cmusco@cs.umass.edu](cmusco@cs.umass.edu)  \narXiv :2607 .05803v 1 [ cs .IR] 7 Jul 2026  \nAbstract  \nLate-interaction retrieval models that use the MaxSim similarity function have shown strong empirical performance, often outperforming single-vector dense and sparse retrieval models. Despite these empirical findings, little is known about the theoretical representation power of MaxSim and how it compares to other retrieval approaches. This paper shows by construction that MaxSim similarity can exactly replicate the inner product between any two non-negative k-sparse vectors with possibly infinite dimension, requiring only O (k) representation space.  \nMoreover, there exist similarities that MaxSim can express while standard vector inner products with the same representation space cannot. Leveraging our theoretical framework, we introduce Signed MaxSim which allows late-interaction models to exactly replicate any real-valued inner product, something we prove standard MaxSim is not capable of. We also show that MaxSim can act as an aggregation of soft-OR operations and as an evaluator of logical expressions in positive Conjunctive Normal Form. Our findings show that MaxSim is at least as capable as standard vector inner products for any non-negative vectors and our extension, Signed MaxSim, is as capable for any vectors. Both similarities possess additional capabilities that inner product cannot replicate, marking one of the first theoretical justifications and quantifications of late-interaction methods. Our theoretical findings are supported empirically: on a retrieval task featuring queries with negations, Signed MaxSim improves out-of-domain performance significantly over a standard ColBERT/MaxSim baseline with nDCG@10 increasing from 0.597 to  \n1.000 under a vocabulary shift and from 0.008 to 0.788 on negation-only queries.  \n1 Introduction  \nThe landscape of neural information retrieval is currently dominated by two primary modeling paradigms. The first, comprising both dense retrievers (e.g., DPR [13]) and learned sparse retrievers (e.g., SNRM [33]), encodes queries and documents into single, fixed-dimensional vectors. Despite their differences in sparsity, both rely on a simple inner product to estimate relevance. The second paradigm, exemplified by late-interaction models like ColBERT [14], represents texts as sets of embeddings and estimates relevance with more complex similarity measures, most often MaxSim (Chamfer Similarity)–a sum of maximum similarities between query and document embeddings.  \nPreprint.  \nDefinition 1.1 (MaxSim Similarity). The MaxSim similarity S : U × V → R, where U , V ⊂ Rn , is defined as:  \nS (U , V) = X mt⟨m, t⟩ .  \nm∈U  \nHere, ⟨· , ·⟩ is the inner product.  \nGenerally speaking, late-interaction models have empirically demonstrated superior performance, particularly in out-of-domain settings [28], but the underlying mechanism for this gap has not been fully elucidated. Empirical work has shown that this gap is not explained by the additional representation space leveraged by late-interaction models [14, 17], as providing more embedding dimensions to single-vector approaches has diminishing returns and does not match the performance of late-interaction models. These results suggest that it is the MaxSim similarity that is the main differentiator. In this work, we theoretically prove this hypothesis, showing that MaxSim simila","cbCaicAZQMDFCwNr","https://ap.wps.com/l/cbCaicAZQMDFCwNr","pdf",460016,7,1,21,"English","en",105,"# Abstract\n# Introduction\n## MaxSim Subsumes Inner-Product Similarity\n## MaxSim’s Limits for Real-Valued Vectors\n## Signed MaxSim and Exact Inner-Product Replication\n## Connections to Sparse Inner Products and Boolean Logic\n## Experimental Support","[{\"question\":\"What does this paper prove about MaxSim’s ability to represent similarity functions?\",\"answer\":\"It proves by construction that MaxSim can exactly replicate the inner product between any two non-negative k-sparse vectors using only O(k) representation space. It also shows there are similarities MaxSim can express that inner products with the same representation space cannot.\"},{\"question\":\"What is Signed MaxSim and what limitation does it address?\",\"answer\":\"Signed MaxSim is an extension that enables late-interaction models to exactly replicate any real-valued inner product. The paper proves standard MaxSim cannot achieve complete parity with inner-product replication for general real-valued vectors under fixed embedding constraints.\"},{\"question\":\"How do the theoretical findings relate to empirical retrieval performance?\",\"answer\":\"The paper supports the theory experimentally using a retrieval task with queries containing negations. Signed MaxSim improves out-of-domain performance compared with a standard ColBERT/MaxSim baseline, with reported nDCG@10 gains under vocabulary shift and on negation-only queries.\"}]",1784191995,53,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"quantifying-and-expanding-the-theoretical-capacity-of-late-interaction-retrieval-models","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/quantifying-and-expanding-the-theoretical-capacity-of-late-interaction-retrieval-models/84010/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What does this paper prove about MaxSim’s ability to represent similarity functions?","Question",{"text":76,"@type":77},"It proves by construction that MaxSim can exactly replicate the inner product between any two non-negative k-sparse vectors using only O(k) representation space. It also shows there are similarities MaxSim can express that inner products with the same representation space cannot.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is Signed MaxSim and what limitation does it address?",{"text":81,"@type":77},"Signed MaxSim is an extension that enables late-interaction models to exactly replicate any real-valued inner product. The paper proves standard MaxSim cannot achieve complete parity with inner-product replication for general real-valued vectors under fixed embedding constraints.",{"name":83,"@type":74,"acceptedAnswer":84},"How do the theoretical findings relate to empirical retrieval performance?",{"text":85,"@type":77},"The paper supports the theory experimentally using a retrieval task with queries containing negations. Signed MaxSim improves out-of-domain performance compared with a standard ColBERT/MaxSim baseline, with reported nDCG@10 gains under vocabulary shift and on negation-only queries.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":46,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},"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":107,"slug":138},19,"General","general"]