Almost-sharp $O(k^{-1} \log k)$ convergence rate for the Sinkhorn algorithm in the asymptotically scalable case

Guillaume Wang

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The Sinkhorn algorithm converges at a rate of O(k^{-1} log k) in ℓ1-norm marginal error for asymptotically scalable problems.

2026-07-01 09:03 UTC pith:TH6UZUYX

load-bearing objection Wang shows an O(k^{-1} log k) rate for Sinkhorn under the asymptotically scalable assumption, nearly closing the gap to the lower bound.

arxiv 2604.26265 v3 pith:TH6UZUYX submitted 2026-04-29 math.OC

Almost-sharp O(k⁻¹ log k) convergence rate for the Sinkhorn algorithm in the asymptotically scalable case

Guillaume Wang This is my paper

classification math.OC

keywords Sinkhorn algorithmconvergence rateoptimal transportmatrix scalingasymptotically scalable casemarginal errorℓ1 norm

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core 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. 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.

What carries the argument

The generalization of the positive-entry argument to the asymptotically scalable case, which supplies the improved rate bound.

Load-bearing premise

The problem instance must belong to the asymptotically scalable case.

What would settle it

An explicit asymptotically scalable instance on which the ℓ1 marginal error remains larger than C k^{-1} log k for arbitrarily large k.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Wang shows an O(k^{-1} log k) rate for Sinkhorn under the asymptotically scalable assumption, nearly closing the gap to the lower bound.

read the 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.

The 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.

The 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.

The citation pattern looks standard for the subfield and does not appear to omit key references. No circularity is visible from the stated argument.

This 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.

I 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.

Referee Report

0 major / 1 minor

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).

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.

minor comments (1)

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.

Simulated Author's Rebuttal

0 responses · 0 unresolved

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.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives an O(k^{-1} log k) rate for Sinkhorn under the asymptotically scalable case by generalizing the positive-entry analysis of Dvurechensky et al. (2018), an independent prior result. No self-citations appear in the load-bearing steps, no parameters are fitted and then renamed as predictions, and the structural premise (asymptotically scalable case) is an explicit external assumption rather than a self-definition. The derivation chain is therefore self-contained against the cited external benchmark and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a theoretical convergence proof; the ledger consists of standard background results in optimal transport and analysis.

axioms (1)

standard math Standard properties of Sinkhorn iterations and marginal error measures in nonnegative matrix scaling.
The result generalizes the positive case analyzed by Dvurechensky et al. (2018).

pith-pipeline@v0.9.1-grok · 5613 in / 1021 out tokens · 33687 ms · 2026-07-01T09:03:39.876870+00:00 · methodology

0 comments

read the original abstract

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).

Review history (3 revisions) →

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages · 1 internal anchor

[1]

arXiv preprint arXiv:2011.12823 , year=

[Ach93] Eva Achilles. “Implications of convergence rates in Sinkhorn balancing”. In:Linear algebra and its applications187 (1993), pp. 109–112. [ALOW17] Zeyuan Allen-Zhu, Yuanzhi Li, Rafael Oliveira, and Avi Wigderson. “Much faster algorithms for matrix scaling”. In:2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS). IEEE. 2017, pp....

work page arXiv 1993
[2]

A review of matrix scaling and Sinkhorn's normal form for matrices and positive maps

Cambridge University Press. 1995, pp. 31–55. [FL89] Joel Franklin and Jens Lorenz. “On the scaling of multidimensional matrices”. In: Linear Algebra and its applications114 (1989), pp. 717–735. [GN25] Promit Ghosal and Marcel Nutz. “On the convergence rate of Sinkhorn’s algorithm”. In:Mathematics of Operations Research(2025). [HHS24] Koyo Hayashi, Hiroshi...

work page internal anchor Pith review Pith/arXiv arXiv 1995
[3]

On Sinkhorn’s algorithm and choice modeling

[QGGU25] Zhaonan Qu, Alfred Galichon, Wenzhi Gao, and Johan Ugander. “On Sinkhorn’s algorithm and choice modeling”. In:Operations Research(2025). [SK67] Richard Sinkhorn and Paul Knopp. “Concerning nonnegative matrices and doubly stochastic matrices”. In:Pacific Journal of Mathematics21.2 (1967), pp. 343–348. [Sou91] George W Soules. “The rate of converge...

work page 2025

[1] [1]

arXiv preprint arXiv:2011.12823 , year=

[Ach93] Eva Achilles. “Implications of convergence rates in Sinkhorn balancing”. In:Linear algebra and its applications187 (1993), pp. 109–112. [ALOW17] Zeyuan Allen-Zhu, Yuanzhi Li, Rafael Oliveira, and Avi Wigderson. “Much faster algorithms for matrix scaling”. In:2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS). IEEE. 2017, pp....

work page arXiv 1993

[2] [2]

A review of matrix scaling and Sinkhorn's normal form for matrices and positive maps

Cambridge University Press. 1995, pp. 31–55. [FL89] Joel Franklin and Jens Lorenz. “On the scaling of multidimensional matrices”. In: Linear Algebra and its applications114 (1989), pp. 717–735. [GN25] Promit Ghosal and Marcel Nutz. “On the convergence rate of Sinkhorn’s algorithm”. In:Mathematics of Operations Research(2025). [HHS24] Koyo Hayashi, Hiroshi...

work page internal anchor Pith review Pith/arXiv arXiv 1995

[3] [3]

On Sinkhorn’s algorithm and choice modeling

[QGGU25] Zhaonan Qu, Alfred Galichon, Wenzhi Gao, and Johan Ugander. “On Sinkhorn’s algorithm and choice modeling”. In:Operations Research(2025). [SK67] Richard Sinkhorn and Paul Knopp. “Concerning nonnegative matrices and doubly stochastic matrices”. In:Pacific Journal of Mathematics21.2 (1967), pp. 343–348. [Sou91] George W Soules. “The rate of converge...

work page 2025