{"paper":{"title":"Resolving Matrix Spencer Conjecture Up to Poly-logarithmic Rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DM","math.CO"],"primary_cat":"cs.DS","authors_text":"Haotian Jiang, Nikhil Bansal, Raghu Meka","submitted_at":"2022-08-24T03:42:44Z","abstract_excerpt":"We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric $d \\times d$ matrices $A_1,\\ldots,A_n$ each with $\\|A_i\\|_{\\mathsf{op}} \\leq 1$ and rank at most $n/\\log^3 n$, one can efficiently find $\\pm 1$ signs $x_1,\\ldots,x_n$ such that their signed sum has spectral norm $\\|\\sum_{i=1}^n x_i A_i\\|_{\\mathsf{op}} = O(\\sqrt{n})$. This result also implies a $\\log n - \\Omega( \\log \\log n)$ qubit lower bound for quantum random access codes encoding $n$ classical bits with advantage $\\gg 1/\\sqrt{n}$.\n  Our proof uses the recent refinement of the non-commutative"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.11286","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/2208.11286/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"}