{"paper":{"title":"Inapproximability of Matrix $p\\rightarrow q$ Norms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Euiwoong Lee, Madhur Tulsiani, Mrinalkanti Ghosh, Venkatesan Guruswami, Vijay Bhattiprolu","submitted_at":"2018-02-21T05:02:10Z","abstract_excerpt":"We study the problem of computing the $p\\rightarrow q$ norm of a matrix $A \\in R^{m \\times n}$, defined as \\[ \\|A\\|_{p\\rightarrow q} ~:=~ \\max_{x \\,\\in\\, R^n \\setminus \\{0\\}} \\frac{\\|Ax\\|_q}{\\|x\\|_p} \\] This problem generalizes the spectral norm of a matrix ($p=q=2$) and the Grothendieck problem ($p=\\infty$, $q=1$), and has been widely studied in various regimes. When $p \\geq q$, the problem exhibits a dichotomy: constant factor approximation algorithms are known if $2 \\in [q,p]$, and the problem is hard to approximate within almost polynomial factors when $2 \\notin [q,p]$.\n  The regime when $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.07425","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"}