{"id":"0d5c3a12-2733-4cf7-89f6-f66747c9c7f2","arxiv_id":"2604.26265","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves almost-sharp O(k^{-1} log k) convergence rate for Sinkhorn algorithm in the asymptotically scalable case.","lead":"The paper proves the Sinkhorn algorithm converges at rate O(k^{-1} log k) in l1-norm marginal error for the asymptotically scalable case. A smart generalist might read it for tighter guarantees on an algorithm used in optimal transport and matrix scaling problems.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the key structural premise that enables the improved rate. With the claim positioned as a direct generalization and no contradictory elements visible, the assessment requires no adjustment.","tokens_in":1591,"tokens_out":262,"duration_ms":18510,"concrete_test":"Extract the precise statement of the main theorem (including the definition of asymptotically scalable) and verify that the proof steps reduce exactly to the positive case when all entries are bounded away from zero; confirm the log k factor arises only from the stated iteration analysis and not from an uncontrolled instance-dependent constant.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim asserts an O(k^{-1} log k) rate in ℓ1 marginal error specifically under the asymptotically scalable case, generalizing the positive-entry argument of Dvurechensky et al. (2018) while sitting between the Ω(k^{-1}) lower bound and prior O(k^{-1/2}) upper bound. The structural premise (asymptotically scalable case) is explicitly required for the argument to go through, and no internal inconsistency, hidden assumption in the rate derivation, or failure of the generalization step is identifiable from the provided abstract and claim description.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves that the Sinkhorn algorithm converges at a rate of O(k^{-1} log k) in ℓ1-norm marginal error in the asymptotically scalable case. This generalizes the positive-entry analysis of Dvurechensky et al. (2018), improves on the prior O(k^{-1/2}) upper bound of Léger (2021), and nearly closes the gap to the Ω(k^{-1}) lower bound of Qu et al. (2025).","tokens_in":1667,"tokens_out":278,"duration_ms":15549,"significance":"If the central derivation holds, the result would be a meaningful advance in the convergence theory of Sinkhorn's algorithm for entropic optimal transport. It supplies a near-optimal rate under a natural structural assumption that strictly contains the positive-matrix case, and the generalization step from the earlier positive-entry argument is the key technical contribution.","major_comments":[],"minor_comments":[{"comment":"The abstract states the rate but does not indicate where in the manuscript the definition of the asymptotically scalable case is formalized or how the ℓ1 marginal error is precisely measured.","section":null}],"recommendation":"uncertain","confidential_remarks":"The provided abstract and claim description do not include the full derivation, so soundness cannot be verified beyond the stated claim; this is the sole reason for the uncertain recommendation."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their summary of the manuscript and for noting its potential significance as a near-optimal convergence result under the asymptotically scalable assumption. No major comments were raised in the report.","responses":[],"tokens_in":1081,"tokens_out":56,"duration_ms":11606,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper establishes an O(k^{-1} log k) upper bound on the l1 marginal error for the Sinkhorn algorithm when the instance is asymptotically scalable. This improves on the prior O(k^{-1/2}) bound from Léger and sits close to the Omega(k^{-1}) lower bound from Qu et al., while extending the positive-entries argument of Dvurechensky et al. to this setting.\n\nThe work is straightforward in its contribution: it identifies the structural condition that lets the earlier analysis carry over and then carries it over. The abstract states the claim and the comparison to prior bounds without overstatement.\n\nThe main limitation is the restriction to the asymptotically scalable case. Outside that regime the paper does not claim improvement, which is honest but narrows the practical reach. Because the full derivation is not reproduced in the material I have, I cannot inspect the technical steps that justify the log k factor or confirm there are no hidden dependencies on the scaling parameters.\n\nThe citation pattern looks standard for the subfield and does not appear to omit key references. No circularity is visible from the stated argument.\n\nThis is for researchers working on convergence rates for matrix scaling and optimal transport algorithms. A reader who already knows the positive-case analysis will see the incremental step clearly. The result is narrow enough that it may not change day-to-day use of Sinkhorn, but the rate improvement is concrete.\n\nI would send it to peer review. The claim is specific and the gap it targets is real; a referee can check the generalization once the full proof is available.","headline":"Wang shows an O(k^{-1} log k) rate for Sinkhorn under the asymptotically scalable assumption, nearly closing the gap to the lower bound.","tokens_in":2141,"tokens_out":398,"would_cite":false,"duration_ms":17272,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The Sinkhorn algorithm converges at a rate of O(k^{-1} log k) in ℓ1-norm marginal error for asymptotically scalable problems.","keywords":["Sinkhorn algorithm","convergence rate","optimal transport","matrix scaling","asymptotically scalable case","marginal error","ℓ1 norm"],"falsifier":"An explicit asymptotically scalable instance on which the ℓ1 marginal error remains larger than C k^{-1} log k for arbitrarily large k.","tokens_in":2481,"feed_emoji":"","tokens_out":569,"duration_ms":18152,"temperature":0.7,"pith_summary":"The paper establishes that the Sinkhorn algorithm reaches an almost-sharp convergence rate of O(k^{-1} log k) measured in ℓ1-norm marginal error when the input lies in the asymptotically scalable case. This improves on the earlier O(k^{-1/2}) upper bound and nearly matches the known Ω(k^{-1}) lower bound. The proof extends an earlier positive-entry argument to this broader structural setting. A reader would care because the faster guarantee directly affects how quickly the algorithm can be certified to produce accurate marginals in large-scale matrix scaling and optimal transport tasks.","feed_headline":"Sinkhorn hits O(k^{-1} log k) rate in scalable case","feed_subtitle":"This nearly matches the known Omega(k^{-1}) lower bound for the asymptotically scalable regime.","key_machinery":"The generalization of the positive-entry argument to the asymptotically scalable case, which supplies the improved rate bound.","core_discovery":"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. This almost closes the gap between the lower bound Ω(k^{-1}) and the previously best known upper bound O(k^{-1/2}), and generalizes the analysis for the positive case.","pith_inferences":["The same proof technique might be adaptable to other scaling algorithms or to unbalanced optimal transport.","Practical implementations could use the new rate to set tighter stopping tolerances once asymptotic scalability is verified.","It remains open whether the logarithmic factor can be removed while staying inside the asymptotically scalable case."],"forward_implications":["The remaining gap to the lower bound is only a logarithmic factor.","The improved rate applies to all problems in the asymptotically scalable regime rather than only strictly positive matrices.","Standard analyses that work in the fully general case are limited to the slower O(k^{-1/2}) rate."],"fun_headline_variants":["Sinkhorn at O(k^{-1} log k) rate in scalable case","O(k^{-1} log k) Sinkhorn convergence asymptotically scalable","Sinkhorn reaches O(k^{-1} log k) in scalable regime","Proven O(k^{-1} log k) rate for Sinkhorn scalable case"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The problem instance must belong to the asymptotically scalable case.","fun_headline_variants_meta":{"raw":{"variants":["Sinkhorn at O(k^{-1} log k) rate in scalable case","O(k^{-1} log k) Sinkhorn convergence asymptotically scalable","Sinkhorn reaches O(k^{-1} log k) in scalable regime","Proven O(k^{-1} log k) rate for Sinkhorn scalable case"]},"model":"grok-4.3","cost_usd":0.004997,"raw_usage":{"total_tokens":2375,"prompt_tokens":537,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":49974500,"prompt_tokens_details":{"text_tokens":537,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1759,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":537,"tokens_out":79,"duration_ms":14659,"temperature":1.0,"reasoning_tokens":1759,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T09:03:39.876870+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit asymptotically scalable instance on which the ℓ1 marginal error remains larger than C k^{-1} log k for arbitrarily large k.","supporting_citations":[],"review_version":3}