{"paper":{"title":"The Paulsen Problem, Continuous Operator Scaling, and Smoothed Analysis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.OA","math.OC","quant-ph"],"primary_cat":"cs.DS","authors_text":"Akshay Ramachandran, Lap Chi Lau, Tsz Chiu Kwok, Yin Tat Lee","submitted_at":"2017-10-06T21:09:34Z","abstract_excerpt":"The Paulsen problem is a basic open problem in operator theory: Given vectors $u_1, \\ldots, u_n \\in \\mathbb R^d$ that are $\\epsilon$-nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors $v_1, \\ldots, v_n \\in \\mathbb R^d$ that exactly satisfy the Parseval's condition and the equal norm condition? Given $u_1, \\ldots, u_n$, the squared distance (to the set of exact solutions) is defined as $\\inf_{v} \\sum_{i=1}^n \\| u_i - v_i \\|_2^2$ where the infimum is over the set of exact solutions. Previous results show that the squared distance of any $\\eps"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.02587","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}