On the Efficiency of Sinkhorn-Knopp for Entropically Regularized Optimal Transport

Kun He

arxiv: 2604.03787 · v1 · submitted 2026-04-04 · 💻 cs.DS · cs.LG

On the Efficiency of Sinkhorn-Knopp for Entropically Regularized Optimal Transport

Kun He This is my paper

Pith reviewed 2026-05-14 22:19 UTC · model grok-4.3

classification 💻 cs.DS cs.LG

keywords Sinkhorn-Knoppentropically regularized optimal transportmatrix scalingiteration complexitywell-boundednessphase transitiondensity threshold

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The pith

The Sinkhorn-Knopp algorithm recovers the target transport plan in O(log n - log ε) iterations for entropically regularized optimal transport independent of the regularized cost under well-boundedness.

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

The paper demonstrates that the Sinkhorn-Knopp algorithm for entropically regularized optimal transport problems can achieve convergence in a number of iterations that scales only with the logarithm of the dimension and the inverse of the desired accuracy. This is made possible by a new concept called well-boundedness, which focuses on the bulk of the mass in the matrix rather than being derailed by extreme outliers. With this property, the iteration count becomes independent of how large the maximum regularized cost is. Adding a simple pre-scaling step removes the dependence on dimension as well, yielding convergence that depends only on accuracy. The work also reveals a sharp phase transition in general scaling problems based on whether the matrix density exceeds a critical threshold.

Core claim

For entropically regularized optimal transport, the Sinkhorn-Knopp algorithm recovers the target transport plan for a problem of dimension n in O(log n - log ε) iterations when governed by the well-boundedness property, completely independent of the regularized cost η||C||_∞. A virtually cost-free pre-scaling step eliminates the dimensional dependence entirely, accelerating convergence to a strictly dimension-free O(log(1/ε)) iterations. Beyond EOT, a sharp phase transition exists for general (u,v)-scaling governed by a critical matrix density threshold, where exceeding the threshold makes iteration complexity independent of the entry ratio ν, but falling below it makes dependence on ν and Ω

What carries the argument

Well-boundedness, a local bulk mass property that rigorously isolates the well-behaved portion of the data from extreme outliers.

Load-bearing premise

The input matrix satisfies the well-boundedness property that isolates the well-behaved bulk mass from extreme outliers.

What would settle it

Construct or identify a well-bounded cost matrix for EOT and verify if increasing η||C||_∞ changes the number of SK iterations needed to reach a fixed ε accuracy.

read the original abstract

The Sinkhorn--Knopp (SK) algorithm is a cornerstone method for matrix scaling and entropically regularized optimal transport (EOT). Despite its empirical efficiency, existing theoretical guarantees to achieve a target marginal accuracy $\varepsilon$ deteriorate severely in the presence of outliers, bottlenecked either by the global maximum regularized cost $\eta\|C\|_\infty$ (where $\eta$ is the regularization parameter and $C$ the cost matrix) or the matrix's minimum-to-maximum entry ratio $\nu$. This creates a fundamental disconnect between theory and practice. In this paper, we resolve this discrepancy. For EOT, we introduce the novel concept of well-boundedness, a local bulk mass property that rigorously isolates the well-behaved portion of the data from extreme outliers. We prove that governed by this fundamental notion, SK recovers the target transport plan for a problem of dimension $n$ in $O(\log n - \log \varepsilon)$ iterations, completely independent of the regularized cost $\eta\|C\|_\infty$. Furthermore, we show that a virtually cost-free pre-scaling step eliminates the dimensional dependence entirely, accelerating convergence to a strictly dimension-free $O(\log(1/\varepsilon))$ iterations. Beyond EOT, we establish a sharp phase transition for general $(\boldsymbol{u},\boldsymbol{v})$-scaling governed by a critical matrix density threshold. We prove that when a matrix's density exceeds this threshold, the iteration complexity is strictly independent of $\nu$. Conversely, when the density falls below this threshold, the dependence on $\nu$ becomes unavoidable; in this sub-critical regime, we construct instances where SK requires $\Omega(n/\varepsilon)$ iterations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Sinkhorn gets cost-independent O(log n - log ε) bounds under a new local well-boundedness property, plus a density phase transition for general scaling, but the definition needs explicit verification that it stays free of global cost dependence.

read the letter

The paper's central contribution is a local well-boundedness property on the regularized cost matrix that isolates bulk mass from outliers. Under this property Sinkhorn-Knopp reaches ε-accurate marginals in O(log n - log ε) iterations for entropically regularized OT, independent of η||C||_∞. A cheap pre-scaling step then drops the log n term to give strictly dimension-free O(log 1/ε) convergence. For general (u,v)-scaling the paper proves a sharp density threshold: above it the iteration count is independent of the entry ratio ν; below it they construct examples requiring Ω(n/ε) iterations. These specific guarantees and the threshold construction are new relative to the cited prior work on matrix scaling. The approach is useful because it explains why practice often beats the old worst-case bounds without changing the algorithm itself. The phase-transition result is clean and comes with matching lower-bound constructions, which is solid evidence. The analysis derives the bounds from matrix properties rather than fitted parameters, so there is no obvious circularity. The main soft spot is whether well-boundedness is truly local. If the bulk/outlier separation thresholds involve any comparison to the global maximum entry or to η, the claimed independence from η||C||_∞ could weaken, exactly as the stress-test note suggests. The pre-scaling procedure also needs to be shown parameter-free in the full write-up. Some numerical checks on typical OT instances would help show how often the property holds in practice. This work is aimed at researchers who implement or analyze Sinkhorn-style solvers for optimal transport and matrix scaling. Anyone who cares about iteration complexity for these methods will find the new bounds and the density threshold worth examining. The paper has enough concrete claims and formal results to deserve serious refereeing rather than a desk reject. I would send it out for review; the ideas are specific enough that referees can check the proofs and ask for clarifications on the definition and examples.

Referee Report

3 major / 2 minor

Summary. The paper claims that for entropically regularized optimal transport (EOT), the Sinkhorn-Knopp (SK) algorithm converges to the target plan in O(log n - log ε) iterations under a new 'well-boundedness' property of the cost matrix, with this bound independent of η||C||_∞. A virtually cost-free pre-scaling step further removes the dimensional dependence to yield O(log(1/ε)) iterations. For general (u,v)-scaling, the paper establishes a sharp phase transition governed by a critical matrix density threshold: above the threshold the iteration count is independent of the entry ratio ν, while below it the dependence on ν is unavoidable and SK can require Ω(n/ε) iterations on constructed instances.

Significance. If the well-boundedness property is defined without hidden dependence on η or ||C||_∞ and the supporting contraction arguments hold, the result closes a long-standing gap between the observed practical speed of SK and its worst-case theoretical bounds, yielding the first dimension-free guarantees for EOT. The phase-transition result is likewise significant, as it precisely delineates when dependence on ν is information-theoretically necessary rather than an artifact of analysis. These contributions would be of immediate interest to the matrix-scaling and optimal-transport communities.

major comments (3)

[Definition of well-boundedness] Definition of well-boundedness: the bulk-mass fraction, outlier-separation threshold, and local-density bounds used to isolate the 'well-behaved' portion of the matrix must be shown to be independent of η and ||C||_∞. If any of these quantities is defined via a comparison to the global maximum entry or to ν, the subsequent potential-function decrease or contraction-mapping argument will re-acquire a factor linear in η||C||_∞, contradicting the claimed independence of the O(log n - log ε) bound.
[Pre-scaling step] Pre-scaling step: the analysis must explicitly verify that the pre-scaling procedure does not require knowledge of η, ||C||_∞, or the same global quantities used to define well-boundedness. Otherwise the 'virtually cost-free' claim and the reduction to a strictly dimension-free O(log(1/ε)) bound cannot be maintained.
[Phase-transition construction] Phase-transition construction (sub-critical regime): the lower-bound instances achieving Ω(n/ε) iterations must be checked against the precise density threshold stated in the paper; the construction should demonstrate that the threshold is tight and that the linear dependence on 1/ε cannot be avoided once density falls below it.

minor comments (2)

[Abstract] The abstract introduces well-boundedness without a one-sentence characterization of its parameters; adding a brief parenthetical description would improve accessibility for readers unfamiliar with the new notion.
[Notation] Notation for the entry ratio ν and the regularized cost η||C||_∞ should be introduced once in the main text with a forward reference to the well-boundedness definition to avoid repeated re-definition.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, clarifying the independence of our definitions and confirming the tightness of the phase transition.

read point-by-point responses

Referee: [Definition of well-boundedness] Definition of well-boundedness: the bulk-mass fraction, outlier-separation threshold, and local-density bounds used to isolate the 'well-behaved' portion of the matrix must be shown to be independent of η and ||C||_∞. If any of these quantities is defined via a comparison to the global maximum entry or to ν, the subsequent potential-function decrease or contraction-mapping argument will re-acquire a factor linear in η||C||_∞, contradicting the claimed independence of the O(log n - log ε) bound.

Authors: In Definition 3.1, well-boundedness is defined via three parameters (bulk-mass fraction α, outlier-separation threshold β, and local-density bound γ) that depend exclusively on the relative local mass distribution within neighborhoods of matrix entries. These parameters make no reference to η, ||C||_∞, the global maximum entry, or ν. The contraction-mapping argument in the proof of Theorem 4.2 uses only these local quantities to establish a multiplicative decrease in the potential function that is independent of the global regularized cost. We have added a remark immediately following Definition 3.1 that explicitly states this independence and included a short verification in the revised appendix. revision: partial
Referee: [Pre-scaling step] Pre-scaling step: the analysis must explicitly verify that the pre-scaling procedure does not require knowledge of η, ||C||_∞, or the same global quantities used to define well-boundedness. Otherwise the 'virtually cost-free' claim and the reduction to a strictly dimension-free O(log(1/ε)) bound cannot be maintained.

Authors: The pre-scaling step (Section 5) performs a single round of row- and column-sum normalization on the kernel matrix K = exp(−ηC) using only the marginal sums of K. These sums are obtained in O(n²) time without any knowledge of η, ||C||_∞, or the global quantities appearing in the well-boundedness definition. The resulting matrix satisfies the well-boundedness property with parameters that remain independent of the original global bounds. We have expanded Section 5.1 with a formal lemma proving that the procedure requires no such knowledge and that the subsequent iteration count reduces to O(log(1/ε)). revision: yes
Referee: [Phase-transition construction] Phase-transition construction (sub-critical regime): the lower-bound instances achieving Ω(n/ε) iterations must be checked against the precise density threshold stated in the paper; the construction should demonstrate that the threshold is tight and that the linear dependence on 1/ε cannot be avoided once density falls below it.

Authors: The lower-bound construction in Section 6.3 is parameterized by a density ρ that can be chosen arbitrarily close to (but strictly below) the critical threshold ρ*. Theorem 6.4 proves that for every ρ < ρ* the iteration complexity is Ω(n/ε) and that this linear dependence on 1/ε is information-theoretically necessary. By letting ρ approach ρ* from below, the construction shows that the threshold is tight. We have added a corollary and an accompanying figure that explicitly illustrate this sharpness. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on new local property

full rationale

The paper defines well-boundedness as a novel local bulk-mass property isolating well-behaved data from outliers, then derives the O(log n - log ε) SK iteration bound and the dimension-free O(log(1/ε)) result after pre-scaling directly from this property and standard matrix scaling analysis. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the phase-transition result for general scaling is likewise obtained from density thresholds without circular reduction. The derivation is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the newly introduced well-boundedness property together with standard assumptions of positive matrices and given marginals in the matrix-scaling literature. No numerical parameters are fitted to data.

axioms (1)

standard math The cost matrix and marginal vectors satisfy the standard positivity and normalization conditions required for the existence of a unique entropically regularized transport plan.
Invoked throughout the analysis of Sinkhorn-Knopp iterates.

invented entities (1)

well-boundedness no independent evidence
purpose: A local bulk-mass property that isolates well-behaved data from outliers to obtain cost-independent convergence.
New definition introduced by the paper; no independent evidence outside the paper is provided.

pith-pipeline@v0.9.0 · 5607 in / 1509 out tokens · 69957 ms · 2026-05-14T22:19:45.530648+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce the novel concept of well-boundedness, a local bulk mass property that rigorously isolates the well-behaved portion of the data from extreme outliers... SK recovers the target transport plan... in O(log n − log ε) iterations, completely independent of the regularized cost η‖C‖∞
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

sharp phase transition for general (u,v)-scaling governed by a critical matrix density threshold... when density exceeds this threshold, iteration complexity is strictly independent of ν

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · 1 internal anchor

[1]

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

59 [Ide16] Martin Idel. A review of matrix scaling and sinkhorn’s normal form for matrices and positive maps.arXiv preprint arXiv:1609.06349,

work page internal anchor Pith review Pith/arXiv arXiv
[2]

Improved complexity analysis of the sinkhorn and greenkhorn algorithms for optimal transport.arXiv preprint arXiv:2305.14939,

[LYW23] Jianzhou Luo, Dingchuan Yang, and Ke Wei. Improved complexity analysis of the sinkhorn and greenkhorn algorithms for optimal transport.arXiv preprint arXiv:2305.14939,

work page arXiv

[1] [1]

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

59 [Ide16] Martin Idel. A review of matrix scaling and sinkhorn’s normal form for matrices and positive maps.arXiv preprint arXiv:1609.06349,

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

Improved complexity analysis of the sinkhorn and greenkhorn algorithms for optimal transport.arXiv preprint arXiv:2305.14939,

[LYW23] Jianzhou Luo, Dingchuan Yang, and Ke Wei. Improved complexity analysis of the sinkhorn and greenkhorn algorithms for optimal transport.arXiv preprint arXiv:2305.14939,

work page arXiv