{"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/unique-powers-of-forms-decompositions-from","title":"Unique powers-of-forms decompositions from simple Gram spectrahedra","arxiv_id":"2305.06860","date":"2023-05-11","proceeding":null,"authors":["Alexander Taveira Blomenhofer"],"abstract":"We consider simultaneous Waring decompositions: Given forms $ f_d $ of degrees $ kd $, $ (d = 2,3 )$, which admit a representation as $ d $-th power sums of $ k $-forms $ q_1,\\ldots,q_m $, when is it possible to reconstruct the addends $ q_1,\\ldots,q_m $ from the power sums $ f_d $? Such powers-of-forms decompositions model the moment problem for mixtures of centered Gaussians. The novel approach of this paper is to use semidefinite programming in order to perform a reduction to tensor decomposition. The proposed method works on typical parameter sets at least as long as $ m\\leq n-1 $, where $ m $ is the rank of the decomposition and $ n $ is the number of variables. While provably not tight, this analysis still gives the currently best known rank threshold for decomposing third order powers-of-forms, improving on previous work in both asymptotics and constant factors. Our algorithm can produce proofs of uniqueness for specific decompositions. A numerical study is conducted on Gaussian random trace-free quadratics, giving evidence that the success probability converges to $ 1 $ in an average case setting, as long as $ m = n $ and $ n\\to \\infty $. Some evidence is given that the algorithm also succeeds on instances of rank $ m = \\Theta(n^2) $.","url_abs":"https://arxiv.org/abs/2305.06860v1","url_pdf":"https://arxiv.org/pdf/2305.06860v1.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":"unique-powers-of-forms-decompositions-from","repo_url":"https://github.com/a44l/powers-of-forms","is_official":1,"mentioned_in_paper":1,"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}