[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81862-en":3,"doc-seo-81862-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},81862,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","Dimension Reduction for Curves: Simplified and Generalized","Random projections are revisited for reducing the dimension of high-dimensional polygonal curves while preserving their continuous Fréchet distance within a multiplicative (1 ± ε) factor. A simplified proof achieves the known O(ε−2 log(nm)) target-dimension bound using sparse oblivious subspace embeddings. Unlike earlier Fréchet-specific methods, the approach extends to distance measures combining maximum, sum, or integral of pairwise Euclidean distances. A generalized curve dissimilarity is defined to subsume Fréchet, q-DTW, and Hausdorff, and the framework is further adapted to piecewise linear surfaces.","arXiv :2607 .03 1 12v 1 [ cs .DS] 3 Jul 2026  \nDimension Reduction for Curves: Simplified and Generalized  \nMatthijs Ebbens \\# University of Cologne, Germany Jie Lu \\#  \nUniversity of Cologne, Germany Alexander Munteanu \\#  \nTU Dortmund, Germany  \n~~ Abstract ~~  \nWe revisit random projections for reducing the dimension of high-dimensional polygonal curves. Drawing from the toolbox of randomized linear algebra, we give a considerably simplified proof of the known O(ε−2 log (nm)) bound on the target dimension of a random projection that preserves the continuous Fréchet distance of polygonal curves up to a factor (1 ± ε) . Our proof is based on the concept of sparse oblivious subspace embeddings. While previous techniques were limited to the case of the Fréchet distance, our techniques are fairly general and extend to all possible distance measures that involve the maximum, a sum or an integral over Euclidean distances between pairs of points on both input curves. We define a generalized dissimilarity measure for curves that includes several popular measures such as Fréchet, q-DTW, Hausdorff, etc. as special cases and show that the same dimension reduction technique works for this generalized dissimilarity measure. Finally, we apply the same framework for dimension reduction to piecewise linear surfaces, after extending the distance measure suitably to such surfaces.  \n2012 ACM Subject Classification Theory of computation → Computational geometry; Theory of computation → Random projections and metric embeddings  \nKeywords and phrases dimension reduction, Fréchet distance, dynamic time warping, polygonal curves, piecewise linear surfaces  \nFunding Matthijs Ebbens: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)– Project Number 459420781 .  \nAlexander Munteanu: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)– Project Number 535889065 .  \n2 Dimension Reduction for Curves: Simplified and Generalized  \n 1  Introduction  \nIn this paper, we study dimension reduction for polygonal curves and build up a fairly general framework, which is widely independent of the dissimilarity measure that is used to compare them. Thus, in contrast to metric embeddings, our framework aims to embed the curves into lower dimensional space, not a specific dissimilarity measure. We first motivate our work from a broader perspective before we return to polygonal curves and surfaces.  \nDimension reduction is one of the most fundamental tasks in computational geometry and has many beneficial consequences. For instance, algorithms for a computational task may have a dependence on the dimension of their input objects, such as points, shapes, or polygonal curves. Reducing the dimension of these objects, say from d to some t ≪ d, allows to apply the same algorithm on the reduced data, but with the dependence on d replaced by the significantly smaller value t. This comes at the cost of a small approximation error which is typically bounded within a factor of (1 ± ε) . Another area where dimension reduction plays a crucial role is the construction of so-called coresets [7, 2] . Coresets aim to reduce the number of data objects rather than their dimension. Classic coreset constructions typically had a small, though polynomial dependence on d, depending on the problem for which they were developed. Sometimes exponential dependencies on the dimension were present or even unavoidable [1, 14 , 31] . More recently, for some problems such as k-means/median clustering, the dependence of the coreset size on the dimension could be completely eliminated [37, 23], which would not have been imaginable without (random) projection techniques such asthe celebrated Johnson-Lindenstrauss (JL) Lemma [25], PCA [26], and their descendants. Extending these techniques to polygonal curves and surfaces is an intriguing open direction, as even the best coresets [16, 17] in this regime suffered from linear dimension dependence.  \n","cbCainuBTNMkhDgM","https://ap.wps.com/l/cbCainuBTNMkhDgM","pdf",554810,1,17,"English","en",105,"# Introduction\n## Results","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper studies dimension reduction for high-dimensional polygonal curves using random projections while preserving important continuous distance measures up to a (1 ± ε) approximation factor.\"},{\"question\":\"How does the paper simplify the proof of the target dimension bound?\",\"answer\":\"It provides a considerably simplified proof of the known O(ε−2 log(nm)) bound by leveraging sparse oblivious subspace embeddings.\"},{\"question\":\"Which distance measures can the generalized framework handle?\",\"answer\":\"The techniques extend beyond Fréchet distance to measures built from maximum, sum, or integrals of Euclidean distances between point pairs on curves, and it introduces a generalized dissimilarity covering Fréchet, q-DTW, Hausdorff, and related 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problem does the paper address?","Question",{"text":74,"@type":75},"The paper studies dimension reduction for high-dimensional polygonal curves using random projections while preserving important continuous distance measures up to a (1 ± ε) approximation factor.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the paper simplify the proof of the target dimension bound?",{"text":79,"@type":75},"It provides a considerably simplified proof of the known O(ε−2 log(nm)) bound by leveraging sparse oblivious subspace embeddings.",{"name":81,"@type":72,"acceptedAnswer":82},"Which distance measures can the generalized framework handle?",{"text":83,"@type":75},"The techniques extend beyond Fréchet distance to measures built from maximum, sum, or integrals of Euclidean distances between point pairs on curves, and it introduces a generalized dissimilarity covering Fréchet, q-DTW, Hausdorff, and related 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