{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:FC4XQUG5IGMFQH4FZ7KEFTQOWL","short_pith_number":"pith:FC4XQUG5","schema_version":"1.0","canonical_sha256":"28b97850dd4198581f85cfd442ce0eb2f5ffc4bf21a8c2adc90dbd9a7e9d1b4a","source":{"kind":"arxiv","id":"2111.03171","version":1},"attestation_state":"computed","paper":{"title":"A New Framework for Matrix Discrepancy: Partial Coloring Bounds via Mirror Descent","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Daniel Dadush, Haotian Jiang, Victor Reis","submitted_at":"2021-11-04T21:44:53Z","abstract_excerpt":"Motivated by the Matrix Spencer conjecture, we study the problem of finding signed sums of matrices with a small matrix norm. A well-known strategy to obtain these signs is to prove, given matrices $A_1, \\dots, A_n \\in \\mathbb{R}^{m \\times m}$, a Gaussian measure lower bound of $2^{-O(n)}$ for a scaling of the discrepancy body $\\{x \\in \\mathbb{R}^n: \\| \\sum_{i=1}^n x_i A_i\\| \\leq 1\\}$. We show this is equivalent to covering its polar with $2^{O(n)}$ translates of the cube $\\frac{1}{n} B^n_\\infty$, and construct such a cover via mirror descent. As applications of our framework, we show:\n  $\\bul"},"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":"2111.03171","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DS","submitted_at":"2021-11-04T21:44:53Z","cross_cats_sorted":[],"title_canon_sha256":"1bfa3c1b3f135253745b54ef5bf2c50901617bb097d595ca010563cf58e88165","abstract_canon_sha256":"5596a7457f37dc21f16f4dfb815f8bce86e5fa83376067674d19db6b9a50f0cf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:29:21.414529Z","signature_b64":"TcAE8IV/Bkaq78YVeTssFUxqtlK4Isao19R2CLVsETCjlBgxjzgNK/KRuqrJIuofu6EtMGoXvOOr0Y+KsVCuDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28b97850dd4198581f85cfd442ce0eb2f5ffc4bf21a8c2adc90dbd9a7e9d1b4a","last_reissued_at":"2026-07-05T03:29:21.414180Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:29:21.414180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A New Framework for Matrix Discrepancy: Partial Coloring Bounds via Mirror Descent","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Daniel Dadush, Haotian Jiang, Victor Reis","submitted_at":"2021-11-04T21:44:53Z","abstract_excerpt":"Motivated by the Matrix Spencer conjecture, we study the problem of finding signed sums of matrices with a small matrix norm. A well-known strategy to obtain these signs is to prove, given matrices $A_1, \\dots, A_n \\in \\mathbb{R}^{m \\times m}$, a Gaussian measure lower bound of $2^{-O(n)}$ for a scaling of the discrepancy body $\\{x \\in \\mathbb{R}^n: \\| \\sum_{i=1}^n x_i A_i\\| \\leq 1\\}$. We show this is equivalent to covering its polar with $2^{O(n)}$ translates of the cube $\\frac{1}{n} B^n_\\infty$, and construct such a cover via mirror descent. As applications of our framework, we show:\n  $\\bul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.03171","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2111.03171/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2111.03171","created_at":"2026-07-05T03:29:21.414244+00:00"},{"alias_kind":"arxiv_version","alias_value":"2111.03171v1","created_at":"2026-07-05T03:29:21.414244+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2111.03171","created_at":"2026-07-05T03:29:21.414244+00:00"},{"alias_kind":"pith_short_12","alias_value":"FC4XQUG5IGMF","created_at":"2026-07-05T03:29:21.414244+00:00"},{"alias_kind":"pith_short_16","alias_value":"FC4XQUG5IGMFQH4F","created_at":"2026-07-05T03:29:21.414244+00:00"},{"alias_kind":"pith_short_8","alias_value":"FC4XQUG5","created_at":"2026-07-05T03:29:21.414244+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2607.08517","citing_title":"Quantum Communication Lower Bounds for Search Problems via Matrix Discrepancy","ref_index":50,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FC4XQUG5IGMFQH4FZ7KEFTQOWL","json":"https://pith.science/pith/FC4XQUG5IGMFQH4FZ7KEFTQOWL.json","graph_json":"https://pith.science/api/pith-number/FC4XQUG5IGMFQH4FZ7KEFTQOWL/graph.json","events_json":"https://pith.science/api/pith-number/FC4XQUG5IGMFQH4FZ7KEFTQOWL/events.json","paper":"https://pith.science/paper/FC4XQUG5"},"agent_actions":{"view_html":"https://pith.science/pith/FC4XQUG5IGMFQH4FZ7KEFTQOWL","download_json":"https://pith.science/pith/FC4XQUG5IGMFQH4FZ7KEFTQOWL.json","view_paper":"https://pith.science/paper/FC4XQUG5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2111.03171&json=true","fetch_graph":"https://pith.science/api/pith-number/FC4XQUG5IGMFQH4FZ7KEFTQOWL/graph.json","fetch_events":"https://pith.science/api/pith-number/FC4XQUG5IGMFQH4FZ7KEFTQOWL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FC4XQUG5IGMFQH4FZ7KEFTQOWL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FC4XQUG5IGMFQH4FZ7KEFTQOWL/action/storage_attestation","attest_author":"https://pith.science/pith/FC4XQUG5IGMFQH4FZ7KEFTQOWL/action/author_attestation","sign_citation":"https://pith.science/pith/FC4XQUG5IGMFQH4FZ7KEFTQOWL/action/citation_signature","submit_replication":"https://pith.science/pith/FC4XQUG5IGMFQH4FZ7KEFTQOWL/action/replication_record"}},"created_at":"2026-07-05T03:29:21.414244+00:00","updated_at":"2026-07-05T03:29:21.414244+00:00"}