{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/strong-coresets-for-subspace-approximation","title":"Dimensionality Reduction for Sum-of-Distances Metric","arxiv_id":"1912.12003","date":"2019-12-27","proceeding":null,"authors":["Zhili Feng","Praneeth Kacham","David P. Woodruff"],"abstract":"We give a dimensionality reduction procedure to approximate the sum of distances of a given set of $n$ points in $R^d$ to any \"shape\" that lies in a $k$-dimensional subspace. Here, by \"shape\" we mean any set of points in $R^d$. Our algorithm takes an input in the form of an $n \\times d$ matrix $A$, where each row of $A$ denotes a data point, and outputs a subspace $P$ of dimension $O(k^{3}/\\epsilon^6)$ such that the projections of each of the $n$ points onto the subspace $P$ and the distances of each of the points to the subspace $P$ are sufficient to obtain an $\\epsilon$-approximation to the sum of distances to any arbitrary shape that lies in a $k$-dimensional subspace of $R^d$. These include important problems such as $k$-median, $k$-subspace approximation, and $(j,l)$ subspace clustering with $j \\cdot l \\leq k$. Dimensionality reduction reduces the data storage requirement to $(n+d)k^{3}/\\epsilon^6$ from nnz$(A)$. Here nnz$(A)$ could potentially be as large as $nd$. Our algorithm runs in time nnz$(A)/\\epsilon^2 + (n+d)$poly$(k/\\epsilon)$, up to logarithmic factors. For dense matrices, where nnz$(A) \\approx nd$, we give a faster algorithm, that runs in time $nd + (n+d)$poly$(k/\\epsilon)$ up to logarithmic factors. Our dimensionality reduction algorithm can also be used to obtain poly$(k/\\epsilon)$ size coresets for $k$-median and $(k,1)$-subspace approximation problems in polynomial time.","url_abs":"https://arxiv.org/abs/1912.12003v5","url_pdf":"https://arxiv.org/pdf/1912.12003v5.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"strong-coresets-for-subspace-approximation","repo_url":"https://gitlab.com/praneeth10/dimensionality-reduction-for-sum-of-distances","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}