{"paper":{"title":"Approximating Operator Norms via Generalized Krivine Rounding","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"cs.DS","authors_text":"Euiwoong Lee, Madhur Tulsiani, Mrinalkanti Ghosh, Venkatesan Guruswami, Vijay Bhattiprolu","submitted_at":"2018-04-10T17:46:12Z","abstract_excerpt":"We consider the $(\\ell_p,\\ell_r)$-Grothendieck problem, which seeks to maximize the bilinear form $y^T A x$ for an input matrix $A$ over vectors $x,y$ with $\\|x\\|_p=\\|y\\|_r=1$. The problem is equivalent to computing the $p \\to r^*$ operator norm of $A$. The case $p=r=\\infty$ corresponds to the classical Grothendieck problem. Our main result is an algorithm for arbitrary $p,r \\ge 2$ with approximation ratio $(1+\\epsilon_0)/(\\sinh^{-1}(1)\\cdot \\gamma_{p^*} \\,\\gamma_{r^*})$ for some fixed $\\epsilon_0 \\le 0.00863$. Comparing this with Krivine's approximation ratio of $(\\pi/2)/\\sinh^{-1}(1)$ for th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.03644","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1804.03644/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}