{"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/computing-bouligand-stationary-points","title":"Computing Bouligand stationary points efficiently in low-rank optimization","arxiv_id":"2409.12298","date":"2024-09-18","proceeding":null,"authors":["Guillaume Olikier","P. -A. Absil"],"abstract":"This paper considers the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on the algebraic variety of all $m$-by-$n$ real matrices of rank at most $r$. Several definitions of stationarity exist for this nonconvex problem. Among them, Bouligand stationarity is the strongest necessary condition for local optimality. Only a handful of algorithms generate a sequence in the variety whose accumulation points are provably Bouligand stationary. Among them, the most parsimonious with (truncated) singular value decompositions (SVDs) or eigenvalue decompositions can still require a truncated SVD of a matrix whose rank can be as large as $\\min\\{m, n\\}-r+1$ if the gradient does not have low rank, which is computationally prohibitive in the typical case where $r \\ll \\min\\{m, n\\}$. This paper proposes a first-order algorithm that generates a sequence in the variety whose accumulation points are Bouligand stationary while requiring SVDs of matrices whose smaller dimension is always at most $r$. A standard measure of Bouligand stationarity converges to zero along the bounded subsequences at a rate at least $O(1/\\sqrt{i+1})$, where $i$ is the iteration counter. Furthermore, a rank-increasing scheme based on the proposed algorithm is presented, which can be of interest if the parameter $r$ is potentially overestimated.","url_abs":"https://arxiv.org/abs/2409.12298v1","url_pdf":"https://arxiv.org/pdf/2409.12298v1.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":"computing-bouligand-stationary-points","repo_url":"https://github.com/golikier/apocalypsefreelowrankoptimization","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"computing-bouligand-stationary-points","repo_url":"https://github.com/golikier/bouligandstationaritylowrankoptimization","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"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}