{"paper":{"title":"Almost-sharp $O(k^{-1} \\log k)$ convergence rate for the Sinkhorn algorithm in the asymptotically scalable case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The Sinkhorn algorithm converges at an O(k^{-1} log k) rate in ell_1 marginal error under the asymptotically scalable condition.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Guillaume Wang","submitted_at":"2026-04-29T03:48:58Z","abstract_excerpt":"We prove that the Sinkhorn algorithm converges at a rate of $O(k^{-1} \\log k)$ in $\\ell_1$-norm marginal error, in the asymptotically scalable case. This almost closes the gap between the lower bound $\\Omega(k^{-1})$ (Qu et al., 2025) and the previously best known upper bound $O(k^{-1/2})$ (L\\'eger, 2021), and generalizes the analysis for the positive case by Dvurechensky et al. (2018)."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that the Sinkhorn algorithm converges at a rate of O(k^{-1} log k) in ℓ1-norm marginal error, in the asymptotically scalable case.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis requires the problem to be in the asymptotically scalable case, whose precise definition and verification conditions are not detailed in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Sinkhorn algorithm converges at O(k^{-1} log k) rate in l1-norm marginal error for asymptotically scalable instances, nearly matching the Omega(k^{-1}) lower bound.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Sinkhorn algorithm converges at an O(k^{-1} log k) rate in ell_1 marginal error under the asymptotically scalable condition.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"689a69cf3fee158144717973c86c9c833c773c804a23c398ea2e79813b506bab"},"source":{"id":"2604.26265","kind":"arxiv","version":3},"verdict":{"id":"feb59d09-a159-492f-ba68-337413c68b9e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T07:20:22.218857Z","strongest_claim":"We prove that the Sinkhorn algorithm converges at a rate of O(k^{-1} log k) in ℓ1-norm marginal error, in the asymptotically scalable case.","one_line_summary":"Sinkhorn algorithm converges at O(k^{-1} log k) rate in l1-norm marginal error for asymptotically scalable instances, nearly matching the Omega(k^{-1}) lower bound.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis requires the problem to be in the asymptotically scalable case, whose precise definition and verification conditions are not detailed in the abstract.","pith_extraction_headline":"The Sinkhorn algorithm converges at an O(k^{-1} log k) rate in ell_1 marginal error under the asymptotically scalable condition."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.26265/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T00:39:43.981308Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:22:50.664095Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"1d080667156a60c48796233569ca9422f2e687e48625f193c836749a34039cf5"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"589c41523f29a4c6bf6b11f8be66d22fbaca80dc5e285d4cb5c93c38f0a32887"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}