{"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/fitting-an-ellipsoid-to-random-points","title":"Fitting an ellipsoid to random points: predictions using the replica method","arxiv_id":"2310.01169","date":"2023-10-02","proceeding":null,"authors":["Antoine Maillard","Dmitriy Kunisky"],"abstract":"We consider the problem of fitting a centered ellipsoid to $n$ standard Gaussian random vectors in $\\mathbb{R}^d$, as $n, d \\to \\infty$ with $n/d^2 \\to \\alpha > 0$. It has been conjectured that this problem is, with high probability, satisfiable (SAT; that is, there exists an ellipsoid passing through all $n$ points) for $\\alpha < 1/4$, and unsatisfiable (UNSAT) for $\\alpha > 1/4$. In this work we give a precise analytical argument, based on the non-rigorous replica method of statistical physics, that indeed predicts a SAT/UNSAT transition at $\\alpha = 1/4$, as well as the shape of a typical fitting ellipsoid in the SAT phase (i.e., the lengths of its principal axes). Besides the replica method, our main tool is the dilute limit of extensive-rank \"HCIZ integrals\" of random matrix theory. We further study different explicit algorithmic constructions of the matrix characterizing the ellipsoid. In particular, we show that a procedure based on minimizing its nuclear norm yields a solution in the whole SAT phase. Finally, we characterize the SAT/UNSAT transition for ellipsoid fitting of a large class of rotationally-invariant random vectors. Our work suggests mathematically rigorous ways to analyze fitting ellipsoids to random vectors, which is the topic of a companion work.","url_abs":"https://arxiv.org/abs/2310.01169v2","url_pdf":"https://arxiv.org/pdf/2310.01169v2.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":"fitting-an-ellipsoid-to-random-points","repo_url":"https://github.com/anmaillard/fitting_ellipsoid_replicas","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":"https://app.syntology.ai/?focus=2310.01169","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}