[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82373-en":3,"doc-seo-82373-105":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},82373,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Terminal Dimension Reduction for Time Series with Applications","Terminal embeddings enable dimension reduction by mapping points from an ambient space into a lower-dimensional space while preserving pairwise distances among a given set with limited distortion. They have proven effective for building k-means and k-median coresets using a small weighted subset. However, they do not directly extend to clustering time-series with straight-line interpolation. This work generalizes terminal embeddings to affine line segments, yielding lines-preserving dimension-free coresets for time-series clustering under the Fréchet distance. Experiments compare favorably against PCA and match JL-style embeddings.","arXiv :2607 .09490v 1 [ cs .DS] 10 Jul 2026  \nTerminal Dimension Reduction for Time Series with Applications Alexander Munteanu∗ Matteo Russo† David Saulpic‡ Chris Schwiegelshohn§  \nJuly 13, 2026  \nAbstract  \nTerminal embeddings have emerged as a powerful tool for dimension reduction. Given a set of points P ⊂ Rd , a terminal embedding is a mapping f : Rd → Rt that preserves the pairwise distance between any pair of points p ∈ P and q ∈ Rd up to small distortion under this mapping. Terminal embeddings have been particularly fruitful for constructing k-means and k-median coresets, where the objective is to find a typically weighted subset Ω of P such that for any candidate solution, the cost of the clustering objective on Ω approximates the cost of the clustering objective on P up to small distortion. Unfortunately, these techniques have not been extended to more complicated structures such as clustering time-series data under common straight-line interpolation between measurements. The main issue is that terminal embeddings, arguably the central technique in this line of research, cannot be linear and are thus not immediately suitable to preserve linear structures. In this work, we develop a generalization of terminal embeddings to affine line-segments that overcomes this issue. We showcase their applicability by using our lines-preserving terminal embeddings to obtain the first dimension-free coresets for clustering time-series under the Fréchet distance. The underlying dimension reduction uses Johnson-Lindenstrauss (JL) embeddings, and our experiments indicate that terminal embeddings perform similarly to JL and favorably against PCA for synthetic and real-world time-series, while only terminal embeddings extend pairwise distance preservation to the full ambient space.  \nKeywords: dimension reduction, terminal embeddings, time-series, Fréchet distance, coresets  \n∗ TU Dortmund, Germany  \n†EPFL, Switzerland  \n‡CNRS & Université Paris Cité, IRIF, Paris, France § Aarhus University, Denmark  \nContents  \n1 Introduction 1  \n1.1 Goals and Problem Setting ................................ 2  \n1.2 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3  \n1.3 Our Techniques   4  \n2 Preliminaries 7  \n3 Terminal Embeddings for Polygonal Curves 7  \n4 Experimental Illustration 11  \nA Further Related Work 19  \nB Omitted Proofs of the Main Part 20  \nC Coresets for Clustering Polygonal Curves under the Fréchet Distance 21  \nC.1 Improved Coreset Size Bounds for the Cohen-Addad et al. [2021] Framework . . . . 22  \nC.2 Centroid Sets for the Fréchet Distance . . . . . . . . . . . . . . . . . . . . . . . . . . 23  \nC.3 A Warm Up: Parallel Lines   24  \nC.3. 1 Set Up   24  \nC.4 CGC.3en4.2e1ralCCoConanssstetructionruction . ofof foCr Parall.el. L. ines .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 24  \n26  \n26  \nC.4.2 Proof of Lemma C.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26  \nCCC ...444...345 CDPoeunalttsiitrnnuggctwEionithveroBytfoheg. rT. Co. ag.seet.sh[e. r.:.](e. r.:. N....C .. is.. a.. n.. A.. p..pr.. ox.. i..m..at..e ..C..en.. tr.. o)[ N](e. r.:. N....C .. is.. a.. n.. A.. p..pr.. ox.. i..m..at..e ..C..en.. tr.. o)[....](e. r.:. N....C .. is.. a.. n.. A.. p..pr.. ox.. i..m..at..e ..C..en.. tr.. o)[C](e. r.:. N....C .. is.. a.. n.. A.. p..pr.. ox.. i..m..at..e ..C..en.. tr.. o)[ .. is.. a.. n.. A.. p..pr.. ox.. i..m..at..e ..C..en.. tr.. o](e. r.:. N....C .. is.. a.. n.. A.. p..pr.. ox.. i..m..at..e ..C..en.. tr.. o)..id.. S..et.. . . . . . ... . . . . 27  \n29  \n31  \nD An Improved Coreset Construction Framework via Chaining 31  \nD.1 Outline of the Argument .................................. 31  \nD.2 Preliminary Facts and Considerations on Groups .................... 36  \nD.3 Group Sampling Algorithm ................................ 38  \nD.4 The Cheap Group Doesn’t Matter   38  \nD.5 Coreset Validity ....................................... 40  \nD.6 Group","cbCaiaYystRoXTDX","https://ap.wps.com/l/cbCaiaYystRoXTDX","pdf",776475,4,1,58,"English","en",105,"# Introduction\n## Goals and Problem Setting\n## Our Results\n## Our Techniques\n# Preliminaries\n# Terminal Embeddings for Polygonal Curves\n# Experimental Illustration\n# Further Related Work","[{\"question\":\"What problem do terminal embeddings address in dimension reduction?\",\"answer\":\"They map points to a lower-dimensional space while preserving pairwise distances for points in the given set up to small distortion, enabling efficient clustering-related approximations.\"},{\"question\":\"Why do existing terminal embedding techniques fail for time-series clustering under interpolation?\",\"answer\":\"They generally cannot be linear, so they cannot directly preserve the linear structure created by straight-line interpolation between time-series measurements.\"},{\"question\":\"How does the proposed method extend terminal embeddings to handle time-series?\",\"answer\":\"It generalizes terminal embeddings to affine line-segments, enabling preservation of line-related structure and supporting dimension-free coresets for clustering time-series under the Fréchet distance.\"}]",1784179985,146,{"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},"terminal-dimension-reduction-for-time-series-with-applications","",{"@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/terminal-dimension-reduction-for-time-series-with-applications/82373/",{"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","2026-07-22","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 do terminal embeddings address in dimension reduction?","Question",{"text":75,"@type":76},"They map points to a lower-dimensional space while preserving pairwise distances for points in the given set up to small distortion, enabling efficient clustering-related approximations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why do existing terminal embedding techniques fail for time-series clustering under interpolation?",{"text":80,"@type":76},"They generally cannot be linear, so they cannot directly preserve the linear structure created by straight-line interpolation between time-series measurements.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed method extend terminal embeddings to handle time-series?",{"text":84,"@type":76},"It generalizes terminal embeddings to affine line-segments, enabling preservation of line-related structure and supporting dimension-free coresets for clustering time-series under the Fréchet 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