{"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/on-o-n-algorithms-for-projection-onto-the-top","title":"On $O(n)$ Algorithms for Projection onto the Top-$k$-sum Sublevel Set","arxiv_id":"2310.07224","date":"2023-10-11","proceeding":null,"authors":["Jake Roth","Ying Cui"],"abstract":"The \\emph{top-$k$-sum} operator computes the sum of the largest $k$ components of a given vector. The Euclidean projection onto the top-$k$-sum sublevel set serves as a crucial subroutine in iterative methods to solve composite superquantile optimization problems. In this paper, we introduce a solver that implements two finite-termination algorithms to compute this projection. Both algorithms have $O(n)$ complexity of floating point operations when applied to a sorted $n$-dimensional input vector, where the absorbed constant is \\emph{independent of $k$}. This stands in contrast to an existing grid-search-inspired method that has $O(k(n-k))$ complexity, a partition-based method with $O(n+D\\log D)$ complexity, where $D\\leq n$ is the number of distinct elements in the input vector, and a semismooth Newon method with a finite termination property but unspecified floating point complexity. The improvement of our methods over the first method is significant when $k$ is linearly dependent on $n$, which is frequently encountered in practical superquantile optimization applications. In instances where the input vector is unsorted, an additional cost is incurred to (partially) sort the vector, whereas a full sort of the input vector seems unavoidable for the other two methods. To reduce this cost, we further derive a rigorous procedure that leverages approximate sorting to compute the projection, which is particularly useful when solving a sequence of similar projection problems. Numerical results show that our methods solve problems of scale $n=10^7$ and $k=10^4$ within $0.05$ seconds, whereas the most competitive alternative, the semismooth Newton-based method, takes about $1$ second. The existing grid-search method and Gurobi's QP solver can take from minutes to hours.","url_abs":"https://arxiv.org/abs/2310.07224v2","url_pdf":"https://arxiv.org/pdf/2310.07224v2.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":"on-o-n-algorithms-for-projection-onto-the-top","repo_url":"https://github.com/jacob-roth/top-k-sum","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"on-o-n-algorithms-for-projection-onto-the-top","repo_url":"https://github.com/jacob-roth/superquantile-opt","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}