{"paper":{"title":"Bounds on Dimension Reduction in the Nuclear Norm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Oded Regev, Thomas Vidick","submitted_at":"2019-01-28T01:28:55Z","abstract_excerpt":"$ \\newcommand{\\schs}{\\scriptstyle{\\mathsf{S}}_1} $For all $n \\ge 1$, we give an explicit construction of $m \\times m$ matrices $A_1,\\ldots,A_n$ with $m = 2^{\\lfloor n/2 \\rfloor}$ such that for any $d$ and $d \\times d$ matrices $A'_1,\\ldots,A'_n$ that satisfy \\[ \\|A'_i-A'_j\\|_{\\schs} \\,\\leq\\, \\|A_i-A_j\\|_{\\schs}\\,\\leq\\, (1+\\delta) \\|A'_i-A'_j\\|_{\\schs} \\] for all $i,j\\in\\{1,\\ldots,n\\}$ and small enough $\\delta = O(n^{-c})$, where $c> 0$ is a universal constant, it must be the case that $d \\ge 2^{\\lfloor n/2\\rfloor -1}$. This stands in contrast to the metric theory of commutative $\\ell_p$ spaces"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.09480","kind":"arxiv","version":1},"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"}