Sinkhorn Algorithm for Sequentially Composed Optimal Transports

Kazuki Watanabe; Noboru Isobe

REVIEW 1 major objections 5 minor 14 references

Sinkhorn Algorithm for Sequentially Composed Optimal Transports

T0 review · 1 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper extends the Sinkhorn algorithm to sequentially composed optimal transport and proves exponential convergence in the Hilbert metric, with a near-linear worst-case complexity for two-stage problems.

desk verdict Solid Sinkhorn extension for sequentially composed OT with a real complexity bound; needs an explicit full-support assumption. read the letter →

arxiv 2412.03120 v4 pith:Z2RIVFFT submitted 2024-12-04 cs.DS cs.LGcs.NAmath.NA

Kazuki Watanabe , Noboru Isobe This is my paper · ORCID

classification cs.DScs.LGcs.NAmath.NA MSC 49Q2290C0868Q25

keywords optimaltransportSinkhornalgorithmsequentiallycomposedentropicregularizationHilbertprojectivemetricworst-casecomplexitycompositionality

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that the Sinkhorn iteration—the standard fast approximation algorithm for optimal transport—can be adapted to sequentially composed optimal transport, a problem in which a chain of transportation plans must agree on shared intermediate marginals. Such problems arise in hierarchical planning, and the paper claims that its adaptation converges exponentially, in the Hilbert metric (a scale-invariant distance on positive vectors), to the optimal dual solution from any positive starting point. For the two-stage case, it further proves a worst-case bound: given a tolerance $\delta>0$, the algorithm returns a feasible pair of plans whose total cost is at most $\mathrm{OPT}+\delta$, using $O\!\left(\max(m_1 m_2, m_2 m_3)\, (\max(\|C^{(1)}\|_\infty,\|C^{(2)}\|_\infty))^2\, \log(m_1 (m_2)^2 m_3)\, \delta^{-3}\|C\|_\infty\right)$ arithmetic operations. This is a near-linear improvement over the cubic cost of earlier algorithms for the same problem.

What carries the argument

The engine of the argument is the Hilbert projective metric on the positive orthant, $d_H(u,v)=\log(\max_i u_i/v_i \cdot \max_j v_j/u_j)$, together with a classical contraction inequality for entrywise positive matrices: applying such a matrix contracts the Hilbert metric by a factor $\lambda(A)<1$ that depends only on the matrix. The Sinkhorn updates are built from positive Gibbs kernels $K^{(i)}=\exp(-C^{(i)}/\varepsilon)$, so every update is a composition of such contractions, producing the exponential rates of the convergence theorems. For the complexity analysis the key machinery is an identity showing that the dual objective increases each iteration by an explicit Kullback-Leibler divergence term between the outer marginals and their current approximations, together with a stopping criterion expressed in terms of the distance from the outer marginals to two subdistributions constructed from the current plans; a lemma bounds the number of iterations by $1 + (4/\delta^2) \log(\|K^{(1)}\|_1 \|K^{(2)}\|_1 / K)$. The final $\delta$-optimal feasible plans are obtained by a rounding projection that converts the approximately feasible half-step matrices into exact plans while inflating the cost by at most the stated error.

What would settle it

Take a two-stage problem ($M=2$) and run the Def. 6 iteration with the Thm 23 stopping criterion for a sequence of shrinking tolerances $\delta$, recording the iteration count; the theorem predicts the count grows like $\delta^{-3}$ (up to a log factor), so a clearly different power would refute the complexity claim. Separately, push one cost entry toward a very large value so the corresponding Gibbs entry approaches zero and measure the per-iteration contraction of the Hilbert distance: the analysis predicts the rate stays $D=\max(\lambda(K^{(1)}),\lambda(K^{(2)}))<1$ for all finite costs, so an observed rate of 1 (no decay) marks where the theorem's assumptions fail.

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Extended reading notes

Core claim

The central discovery is that a Sinkhorn-type alternating scaling works for sequentially composed transport if the update order is chosen so that shared-boundary variables are refreshed before the two outer endpoints, and the iteration is analysed in the Hilbert projective metric. Concretely, for a chain of two transports with Gibbs kernels $K^{(i)}=\exp(-C^{(i)}/\varepsilon)$, the update is $w_{n+1}=\sqrt{((K^{(1)})^\top u_n)/(K^{(2)}v_n)}$, $u_{n+1}=a/(K^{(1)}(1/w_{n+1}))$, $v_{n+1}=b/((K^{(2)})^\top w_{n+1})$, with component-wise operations. The paper proves that this iteration drives the Hilbert distance to the optimal dual vectors to zero at rate $D^n$ with $D=\max(\lambda(K^{(1)}),\lambda(K^{(2)}))<1$, and that the matrices built from these vectors satisfy the outer marginal constraints exactly from the first iteration onward. For the two-stage case it identifies a stopping criterion based on the distance from the outer marginals to two product-rule subdistributions, and shows that once the criterion is met, a single standard rounding step yields $\delta$-optimal feasible plans in near-linear worst-case time.

Load-bearing premise

The proofs require every Gibbs kernel $K^{(i)}=\exp(-C^{(i)}/\varepsilon)$ to be entrywise positive, which means every cost-matrix entry must be finite; if any cost is allowed to be infinite (a forbidden transition), the contraction coefficient that drives the exponential convergence and the complexity bound can reach 1 and the analysis collapses.

Editorial extensions

If this is right

At integer iterations the induced plans already match the two outer marginals exactly, so the only constraint violated is the shared-boundary consistency, and the paper shows its violation shrinks exponentially with the iteration count.
For two-stage problems the algorithm returns a feasible pair of plans whose total cost is within $\delta$ of the optimum, with worst-case complexity $O(\max(m_1 m_2, m_2 m_3) (\max(\|C^{(1)}\|_\infty,\|C^{(2)}\|_\infty))^2 \log(m_1 (m_2)^2 m_3) \delta^{-3} \|C\|_\infty)$, which is near-linear in the matrix sizes.
For chains of any length $M\ge 2$ the same alternating update retains exponential Hilbert-metric convergence, so the algorithmic pattern is not limited to the two-stage setting.
The stopping criterion in the complexity theorem is equivalent, up to one extra iteration, to the more natural boundary-mismatch criterion, so practitioners can use the simpler check without changing the guarantees.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the contraction arguments only use that each interface map is a positive contraction, the same update scheme should adapt to other compositional structures such as parallel compositions or trees, an extension the paper leaves implicit.
The $\delta^{-3}$ dependence in the complexity theorem is one factor of $\delta$ worse than the near-linear $\delta^{-2}$ bound for un-composed Sinkhorn iteration; the paper does not investigate whether this extra factor is intrinsic to sequential composition or an artifact of the analysis.
Numerically, the practical iteration count should grow as the contraction coefficients $\lambda(K^{(i)})$ approach 1 when cost matrices have a large dynamic range, so rescaling experiments on cost entries would connect the worst-case analysis to observed convergence speeds.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my take on arXiv:2412.03120. The paper does something genuinely new: it extends the Sinkhorn iteration to sequentially composed OT (SeqOT) and gives the first convergence and complexity analysis for that variant. The algorithm is a natural coordinate-descent on the dual, and the convergence proof via the Hilbert metric is clean. The M=2 complexity result is the real meat: it follows Altschuler et al.'s template but needs a new stopping criterion, and the appendix proof checks out. I verified the key inequalities (Lemmas 17, 19, 20, Theorem 23) and they are internally consistent.

The main soft spot is the zero-marginal gap. Def. 1 allows a and b to have zero entries, but the proofs of Theorems 12, 14, and 23 assume the Sinkhorn iterates and optimal dual vectors stay in R_+^m. If a_j=0, then u(n,1)_j=0 for all n≥1 and the optimal û(1)_j=0, so the Hilbert metric dH(u(n,1), û(1)) is undefined (0/0). Lemma 17's componentwise ratio and Lemma 19's log terms hit the same boundary. This is not fatal; adding a full-support assumption (a,b>0) or an explicit support-restriction argument fixes it. But as stated, the theorems don't cover the full domain of Def. 1.

Other notes: there are no numerical experiments, which is fine for a theory paper but limits the significance claim. The simplified bound in the abstract hides the max(m1m2,m2m3) factor; the theorem itself is precise. The typos are cosmetic.

Overall: a competent theory paper with a new algorithm and a genuine complexity bound. The gap is real but patchable. I'd send it to peer review and ask for a fix to the marginal assumption plus a remark on how zero entries are handled. A reader working on Sinkhorn variants or compositional OT will get value from it.

Referee Report

1 major / 5 minor

Summary. The paper extends the Sinkhorn algorithm to sequentially composed optimal transport (SeqOT), a hierarchical OT variant with adjacency constraints between successive transport plans. The authors derive a log-domain coordinate-descent update (Def. 4 and Def. 6), prove exponential convergence of the iterates in the Hilbert pseudometric for general M and with an improved rate for M = 2 (Thm. 12 and Thm. 14), and give a worst-case arithmetic complexity bound for M = 2 in the style of Altschuler et al. (Thm. 23). The appendices contain full proofs of the dual derivation, the contraction inequalities, and the complexity bound.

Significance. If the main theorems hold, the paper makes a solid contribution by extending Sinkhorn theory beyond vanilla OT to a compositionally constrained variant, with a worst-case complexity that improves on the previous O(m^3) method for SeqOT. The proof strategy is self-contained, uses no fitted parameters, and the key contraction arguments are explicit. The main caveat is that the results are stated for the full simplex, while the proofs require strictly positive marginals; this is a fixable but load-bearing gap.

major comments (1)

[Def. 1, Def. 4, Def. 6, Prop. 3, Thm. 12, Thm. 14, Thm. 23] The problem definition allows arbitrary probability vectors a and b, including entries equal to zero, but the algorithm and the proofs require a,b > 0. In Def. 4, the update for f(n+1,1)_j contains log(a_j), which is -infinity when a_j = 0; equivalently, in Def. 6, u(n+1,1)_j = 0 for all n when a_j = 0. The Hilbert metric of Def. 8 is defined only on positive vectors, so the quantities dH(u(n,1), û(1)) appearing in Thm. 12 and Thm. 14 become 0/0 undefined on such coordinates. Lemma 17's componentwise identity u(n+1,1)/u(n,1) also degenerates, and the dual objective in Prop. 3 contains terms that are not well-defined at zero marginals. Additionally, the strong-duality claim in Prop. 3 relies on Slater's condition, which fails when a feasible plan cannot be strictly positive because of a zero marginal. The theorems are therefore not proven for the full domain stated in Def. 1. The fix is to add an explicit full-support assumption (a,b > 0) to the main results, or to provide a careful limiting extension to boundary marginals with stated conventions for 0 log 0 and 0/0 ratios. This is load-bearing because it determines the domain of validity of every convergence and complexity claim in the paper.

minor comments (5)

[Title page] The title contains a typo: 'Tran sports' should be 'Transports'.
[Def. 6] The denominator in the update for u(n+1,i) is written as 'K(i) 1mi+1 / u(n,i+1)', which is ambiguous; it should be written as K(i) (1_{m_{i+1}} ⊕ u(n,i+1)) or K(i) · (1/u(n,i+1)) componentwise.
[Prop. 13] The definition of d uses dH(P(0,3), P̂(3)), but P(0,3) is not defined in Def. 7; the intended object is u(0,3), consistent with Thm. 12.
[Lemma 19 proof] In the displayed inequality, the factor 'Û(2)_l' appears where the correct expression is 1/Û(3)_l (or an analogous term), given the definitions of P̂(1) and P̂(2) in Cor. 11. The final bound is correct, but the intermediate formula is dimensionally inconsistent as written.
[Thm. 14 proof sketch] The proof sketch states 'dH(u(n+3,1), U(3)) ≤ D· dH(u(n+1,2), U(2))'; the index on the left should almost certainly be u(n+3,3), not u(n+3,1).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: convergence and complexity results are proven from Birkhoff contraction and Altschuler et al., with no fitted parameter or self-citation used as a premise.

full rationale

The derivation chain is self-contained. The Sinkhorn updates in Def. 6 are derived from the dual objective in Prop. 3 by coordinate-ascent equations from ∂L/∂f=0, not from a prior claim about convergence. The contraction argument in Thm. 12/14 uses Birkhoff's Lemma 9 and elementary Hilbert-metric identities (Lemmas 28-30), which are external and parameter-free. The M=2 complexity proof (Lemma 20, Theorem 23) follows the Altschuler et al. template: it bounds the dual gap L(û)-L(u(0)) via Lemma 19, lower-bounds per-iteration dual progress by KL divergences (Lemma 17), applies Pinsker's inequality, and uses the Altschuler et al. feasibility projection (Lemma 21). The tolerance ε and stopping threshold in Thm. 23 are theoretical choices that make a telescoping error bound, not fitted parameters or predictions from data. The citations to Watanabe and Isobe (2024) are confined to defining the problem and to an introductory complexity comparison; no theorem in the paper cites that prior work as a premise, and nothing in the convergence or complexity proof depends on the authors' earlier results. The boundary-case issue with zero marginals noted in the reader's take is a correctness/domain gap, not a circularity: it does not make the conclusion identical to an input assumption.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. Its analysis rests on standard convex duality, Birkhoff's contraction theorem, and Pinsker's inequality, plus the domain assumption of finite costs so the Gibbs kernels are positive.

assumptions (4)

standard math Slater's condition holds for the primal SeqOT problem
Used in Prop. 3 to assert strong duality; satisfied because strictly feasible positive plans exist for finite costs.
standard math Birkhoff's contraction inequality for positive matrices
Lemma 9 is the core contraction tool; requires entrywise positive matrices.
standard math Pinsker's inequality for distributions and subdistributions
Used in Lem. 20 and Prop. 13 to convert KL divergences to L1 errors.
domain assumption All transport costs are finite (cost matrices in R^{≥0})
Def. 1 restricts costs to finite nonnegative values, making Gibbs kernels positive. Infinite costs would break the contraction coefficient bound.

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Cite this review

Pith. "Pith review of Sinkhorn Algorithm for Sequentially Composed Optimal Transports." pith.science (2026). https://pith.science/paper/Z2RIVFFT

@misc{pith2026241203120,
  author       = {Pith},
  title        = {Pith review of: Sinkhorn Algorithm for Sequentially Composed Optimal Transports},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z2RIVFFT}},
  note         = {Machine review of arXiv:2412.03120}
}

read the original abstract

Sinkhorn algorithm is the de-facto standard approximation algorithm for optimal transport, which has been applied to a variety of applications, including image processing and natural language processing. In theory, the proof of its convergence follows from the convergence of the Sinkhorn--Knopp algorithm for the matrix scaling problem, and Altschuler et al. show that its worst-case time complexity is in near-linear time. Very recently, sequentially composed optimal transports were proposed by Watanabe and Isobe as a hierarchical extension of optimal transports. In this paper, we present an efficient approximation algorithm, namely Sinkhorn algorithm for sequentially composed optimal transports, for its entropic regularization. Furthermore, we present a theoretical analysis of the Sinkhorn algorithm, namely (i) its exponential convergence to the optimal solution with respect to the Hilbert pseudometric, and (ii) a worst-case complexity analysis for the case of one sequential composition.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Altschuler, Jonathan Weed, and Philippe Rigollet

Jason M. Altschuler, Jonathan Weed, and Philippe Rigollet. Near-linear time approximation algo- rithms for optimal transport via Sinkhorn iteration. In NIPS, pages 1964–1974,

work page 1964

[2] [2]

We refer Boyd and V andenberghe(2004) as a reference for the Lagrange duality, including the deﬁnition of the Lagrangian

Deﬁnition 26 The Lagrangian L ( ( P(i)) i∈[M ], ( f (i)) i∈[M +1] ) is given by ( M∑ i=1 ⟨ C(i), P(i) ⟩ − ǫH ( P(i) ) ) + ⟨ f (1), a− P(1)1m2 ⟩ + ⟨ f (M +1), b− P(M )⊤ 1mM ⟩ + M∑ i=2 ⟨ f (i), P(i−1)⊤ 1mi−1− P(i)1mi+1 ⟩ . We refer Boyd and V andenberghe(2004) as a reference for the Lagrange duality, including the deﬁnition of the Lagrangian. Lemma 27 Assum...

work page 2004

[3] [8]

Error estimate for regularized optimal transport problems via Bregman divergence

Keiichi Morikuni, Koya Sakakibara, and Asuka Takatsu. Erro r estimate for regularized optimal transport problems via bregman divergence. CoRR, abs/2309.11666,

work page Pith review arXiv

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URL http://projecteuclid.org/euclid.mmj/1028990177

ISSN 0026- 2285,1945-2365. URL http://projecteuclid.org/euclid.mmj/1028990177. Richard Sinkhorn and Paul Knopp. Concerning nonnegative ma trices and doubly stochastic matri- ces. Paciﬁc Journal of Mathematics , 21(2):343–348,

arXiv 1945

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Slivkins et al

P(n, 2)1m3 ) 1/ 2)) The second inequality comes from the equality (x + y)2 + (x− y)2 = 2x2 + 2y2 for any x, y∈ R, and the last inequality comes from the Pinsker’s inequality (e.g. Slivkins et al. (2019)). We note that the Pinsker’s inequality‖P− Q‖2 1≤ 2 KL(P‖ Q) holds even for the case that P is a distribution and Q is a sub-distribution. This is because...

work page 2019

[6] [1957]

doi: 10.2307/19 92971

ISSN 0002-9947,1088-6850. doi: 10.2307/19 92971. URL https://doi.org/10.2307/1992971. Stephen Boyd and Lieven V andenberghe. Convex optimization. Cambridge university press,

doi:10.2307/19

[7] [1989]

doi: https://doi.org/10.1016/0024-3795(89) 90490-4

ISSN 0024-3795. doi: https://doi.org/10.1016/0024-3795(89) 90490-4. URL https://www.sciencedirect.com/science/article/pii/0024379589904904. Special Issue Dedicated to Alan J. Hoffman. Takumi Fukunaga and Hiroyuki Kasai. On the convergence of se mi-relaxed sinkhorn with marginal constraint and OT distance gaps. CoRR, abs/2205.13846,

arXiv

[8] [1998]

URL https://onlinelibrary.wiley.com/doi/abs/10.1002/%28SICI%291097-0312%28199801%29 Giacomo Greco, Maxence Noble, Giovanni Conforti, and Alain Durmus

doi: https://doi.org/10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/%28SICI%291097-0312%28199801%29 Giacomo Greco, Maxence Noble, Giovanni Conforti, and Alain Durmus. Non-asymptotic conver- gence bounds for sinkhorn iterates and their gradients: a co upling approach. In COLT, volume 195 of ...

work page doi:10.1002/(sici)1097-0312(199801)51:1