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We prove that this classical quantity approximates the \\emph{hereditary discrepancy} $\\mathrm{herdisc}\\ A$ as follows: $\\gamma_2(A) = {O(\\log m)}\\cdot \\mathrm{herdisc}\\ A$ and $\\mathrm{herdisc}\\ A = O(\\sqrt{\\log m}\\,)\\cdot\\gamma_2(A) $. Since $\\gamma_2$ is polynomial-time computable, this gives a polynomial-time approximation algorithm for hereditary discrepa"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1408.1376","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-06T18:59:10Z","cross_cats_sorted":["cs.CG","cs.DS"],"title_canon_sha256":"82195e77e97fd4f16a3719562df5ce64aba5cccb3c913341cbddd0da9170a318","abstract_canon_sha256":"0b403e3bd0afbf7608533b9d600f79e79f1e95e6f39eaa421777f43e767db344"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:08.362238Z","signature_b64":"DESYtQiiTBCYJq1zsdybO0dgC2dTbz5Tv/yL806DYDYKiBNVAV5jZ+xeaiw0H5Tye4vR74wU59GVYtkjOqb2Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"61f32514e01aceb8df59a29985769a2fb1cb70bfc0785de542346366d1b1db17","last_reissued_at":"2026-05-18T02:19:08.361810Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:08.361810Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Factorization Norms and Hereditary Discrepancy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DS"],"primary_cat":"math.CO","authors_text":"Aleksandar Nikolov, Jiri Matousek, Kunal Talwar","submitted_at":"2014-08-06T18:59:10Z","abstract_excerpt":"The $\\gamma_2$ norm of a real $m\\times n$ matrix $A$ is the minimum number $t$ such that the column vectors of $A$ are contained in a $0$-centered ellipsoid $E\\subseteq\\mathbb{R}^m$ which in turn is contained in the hypercube $[-t, t]^m$. We prove that this classical quantity approximates the \\emph{hereditary discrepancy} $\\mathrm{herdisc}\\ A$ as follows: $\\gamma_2(A) = {O(\\log m)}\\cdot \\mathrm{herdisc}\\ A$ and $\\mathrm{herdisc}\\ A = O(\\sqrt{\\log m}\\,)\\cdot\\gamma_2(A) $. Since $\\gamma_2$ is polynomial-time computable, this gives a polynomial-time approximation algorithm for hereditary discrepa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.1376","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1408.1376","created_at":"2026-05-18T02:19:08.361877+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.1376v2","created_at":"2026-05-18T02:19:08.361877+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.1376","created_at":"2026-05-18T02:19:08.361877+00:00"},{"alias_kind":"pith_short_12","alias_value":"MHZSKFHADLHL","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_16","alias_value":"MHZSKFHADLHLRX2Z","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_8","alias_value":"MHZSKFHA","created_at":"2026-05-18T12:28:38.356838+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2604.02385","citing_title":"Validity, Sparse Holes, and Breadth in Language Generation: Banach Density, Topology, and Geometry","ref_index":24,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MHZSKFHADLHLRX2ZUKMYK5U2F6","json":"https://pith.science/pith/MHZSKFHADLHLRX2ZUKMYK5U2F6.json","graph_json":"https://pith.science/api/pith-number/MHZSKFHADLHLRX2ZUKMYK5U2F6/graph.json","events_json":"https://pith.science/api/pith-number/MHZSKFHADLHLRX2ZUKMYK5U2F6/events.json","paper":"https://pith.science/paper/MHZSKFHA"},"agent_actions":{"view_html":"https://pith.science/pith/MHZSKFHADLHLRX2ZUKMYK5U2F6","download_json":"https://pith.science/pith/MHZSKFHADLHLRX2ZUKMYK5U2F6.json","view_paper":"https://pith.science/paper/MHZSKFHA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.1376&json=true","fetch_graph":"https://pith.science/api/pith-number/MHZSKFHADLHLRX2ZUKMYK5U2F6/graph.json","fetch_events":"https://pith.science/api/pith-number/MHZSKFHADLHLRX2ZUKMYK5U2F6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MHZSKFHADLHLRX2ZUKMYK5U2F6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MHZSKFHADLHLRX2ZUKMYK5U2F6/action/storage_attestation","attest_author":"https://pith.science/pith/MHZSKFHADLHLRX2ZUKMYK5U2F6/action/author_attestation","sign_citation":"https://pith.science/pith/MHZSKFHADLHLRX2ZUKMYK5U2F6/action/citation_signature","submit_replication":"https://pith.science/pith/MHZSKFHADLHLRX2ZUKMYK5U2F6/action/replication_record"}},"created_at":"2026-05-18T02:19:08.361877+00:00","updated_at":"2026-05-18T02:19:08.361877+00:00"}