{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:5VCTJSKEXNFVXXVPZ54YJ4MPJ4","short_pith_number":"pith:5VCTJSKE","schema_version":"1.0","canonical_sha256":"ed4534c944bb4b5bdeafcf7984f18f4f01a594ca7662882dc2f5a5a8d3089c28","source":{"kind":"arxiv","id":"2607.04388","version":1},"attestation_state":"computed","paper":{"title":"Optimal Online Discrepancy Minimization in Linear Time","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Ishaq Aden-Ali","submitted_at":"2026-07-05T16:31:42Z","abstract_excerpt":"We provide an online algorithm with the following guarantee: for any fixed sequence of vectors $v_1,\\dots,v_T \\in \\mathbf{R}^d$ with $\\|v_i\\|_2\\le 1$, the algorithm assigns each arriving vector $v_t$ a random sign $\\varepsilon_t$ such that every prefix sum $\\sum_{i=1}^t \\varepsilon_i v_i $ can be written as the sum of three coupled standard Gaussian vectors. Our algorithm runs in $O(dT)$ time and achieves the optimal prefix discrepancy bound \\[ \\max_{1 \\le t \\le T}\\left\\| \\sum_{i=1}^t \\varepsilon_i v_i \\right\\|_\\infty = O\\left( \\sqrt{\\log T} \\right), \\] with high probability. This recovers the"},"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":"2607.04388","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DS","submitted_at":"2026-07-05T16:31:42Z","cross_cats_sorted":[],"title_canon_sha256":"645744a03f25ff8bd759ea1e8253040621778827215cd5abaa23aaa0cd017d15","abstract_canon_sha256":"fcbd8ee72b39bf0edbd7be20f14c89046f97531c33db0c67dc86a01d21a77ce5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:19:16.079955Z","signature_b64":"3oXhC/wDM2l3OOL69mF9bT8p5Ht+p5YAGNs2bhjeHk5DmPTVhXqMiUUcsze1apQOf5Z3fUKzfFer/07CoV45Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed4534c944bb4b5bdeafcf7984f18f4f01a594ca7662882dc2f5a5a8d3089c28","last_reissued_at":"2026-07-07T02:19:16.079050Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:19:16.079050Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal Online Discrepancy Minimization in Linear Time","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Ishaq Aden-Ali","submitted_at":"2026-07-05T16:31:42Z","abstract_excerpt":"We provide an online algorithm with the following guarantee: for any fixed sequence of vectors $v_1,\\dots,v_T \\in \\mathbf{R}^d$ with $\\|v_i\\|_2\\le 1$, the algorithm assigns each arriving vector $v_t$ a random sign $\\varepsilon_t$ such that every prefix sum $\\sum_{i=1}^t \\varepsilon_i v_i $ can be written as the sum of three coupled standard Gaussian vectors. Our algorithm runs in $O(dT)$ time and achieves the optimal prefix discrepancy bound \\[ \\max_{1 \\le t \\le T}\\left\\| \\sum_{i=1}^t \\varepsilon_i v_i \\right\\|_\\infty = O\\left( \\sqrt{\\log T} \\right), \\] with high probability. This recovers the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04388","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/2607.04388/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":"2607.04388","created_at":"2026-07-07T02:19:16.079175+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.04388v1","created_at":"2026-07-07T02:19:16.079175+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.04388","created_at":"2026-07-07T02:19:16.079175+00:00"},{"alias_kind":"pith_short_12","alias_value":"5VCTJSKEXNFV","created_at":"2026-07-07T02:19:16.079175+00:00"},{"alias_kind":"pith_short_16","alias_value":"5VCTJSKEXNFVXXVP","created_at":"2026-07-07T02:19:16.079175+00:00"},{"alias_kind":"pith_short_8","alias_value":"5VCTJSKE","created_at":"2026-07-07T02:19:16.079175+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.12490","citing_title":"Online balancing of vectors with small coordinates","ref_index":1,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5VCTJSKEXNFVXXVPZ54YJ4MPJ4","json":"https://pith.science/pith/5VCTJSKEXNFVXXVPZ54YJ4MPJ4.json","graph_json":"https://pith.science/api/pith-number/5VCTJSKEXNFVXXVPZ54YJ4MPJ4/graph.json","events_json":"https://pith.science/api/pith-number/5VCTJSKEXNFVXXVPZ54YJ4MPJ4/events.json","paper":"https://pith.science/paper/5VCTJSKE"},"agent_actions":{"view_html":"https://pith.science/pith/5VCTJSKEXNFVXXVPZ54YJ4MPJ4","download_json":"https://pith.science/pith/5VCTJSKEXNFVXXVPZ54YJ4MPJ4.json","view_paper":"https://pith.science/paper/5VCTJSKE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.04388&json=true","fetch_graph":"https://pith.science/api/pith-number/5VCTJSKEXNFVXXVPZ54YJ4MPJ4/graph.json","fetch_events":"https://pith.science/api/pith-number/5VCTJSKEXNFVXXVPZ54YJ4MPJ4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5VCTJSKEXNFVXXVPZ54YJ4MPJ4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5VCTJSKEXNFVXXVPZ54YJ4MPJ4/action/storage_attestation","attest_author":"https://pith.science/pith/5VCTJSKEXNFVXXVPZ54YJ4MPJ4/action/author_attestation","sign_citation":"https://pith.science/pith/5VCTJSKEXNFVXXVPZ54YJ4MPJ4/action/citation_signature","submit_replication":"https://pith.science/pith/5VCTJSKEXNFVXXVPZ54YJ4MPJ4/action/replication_record"}},"created_at":"2026-07-07T02:19:16.079175+00:00","updated_at":"2026-07-07T02:19:16.079175+00:00"}