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arxiv: 2606.26739 · v1 · pith:7ZYJIOTPnew · submitted 2026-06-25 · 🧮 math.OC

Sinkhorn-Knopp balancing with generalised martingale-type constraints

Abigail Langbridge , Martin Corless , Robert Shorten This is my paper

Pith reviewed 2026-06-26 03:26 UTC · model grok-4.3

classification 🧮 math.OC
keywords Sinkhorn algorithmoptimal transportentropic regularizationmartingale constraintsresource allocationiterative scalingsharing economy
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The pith

Generalized optimal transport problems with martingale-type constraints admit provably convergent Sinkhorn-like algorithms.

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

The paper seeks to extend the standard entropically regularized optimal transport setup so that general transportation constraints and heterogeneous flexibility in the marginal constraints can still be handled by iterative scaling. This matters because many resource distribution tasks, including those in sharing economies and martingale settings, require both a plan and its realizable outcomes under such constraints. If the extension holds, these problems become solvable by adapted Sinkhorn procedures whose convergence is guaranteed when the flexibility conditions are met. The work supplies both the algorithmic construction and examples that demonstrate its practical use.

Core claim

The central claim is that a generalization of the classic entropically regularised Optimal Transport formulation, incorporating general transportation constraints together with heterogeneous flexibility in the marginal constraints, can be solved by provably convergent Sinkhorn-like algorithms obtained through iterative scaling procedures analogous to the standard Sinkhorn-Knopp method.

What carries the argument

Generalized entropically regularised optimal transport problem with martingale-type constraints, solved by iterative scaling that extends the Sinkhorn-Knopp procedure.

If this is right

  • Sharing-economy resource allocation problems can be solved for both the transportation plan and its realisation under the added constraints.
  • Martingale-type problems in optimal transport become computationally tractable within the same entropic framework.
  • The algorithms remain efficient for resource distribution tasks that combine general transportation rules with varying marginal flexibility.
  • Convergence holds under the heterogeneity conditions given for the marginal constraints.

Where Pith is reading between the lines

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

  • The same scaling approach might be tested on dynamic or time-indexed variants of these constraints to check whether convergence persists.
  • Integration with other forms of side constraints, such as capacity limits on routes, could be explored as a direct next step.
  • Numerical comparisons on benchmark sharing-economy instances would quantify any added computational cost relative to the classical Sinkhorn case.

Load-bearing premise

The generalized transportation and marginal constraints admit solutions that can be recovered via iterative scaling procedures analogous to the standard Sinkhorn-Knopp algorithm, with convergence guaranteed under the stated heterogeneity in flexibility.

What would settle it

A concrete small-scale instance with heterogeneous marginal flexibility in which the proposed iterative scaling procedure either diverges or returns a plan that violates one of the stated transportation or martingale constraints.

Figures

Figures reproduced from arXiv: 2606.26739 by Abigail Langbridge, Martin Corless, Robert Shorten.

Figure 1
Figure 1. Figure 1: Consider the distribution of male and female drivers in ˜u, where male drivers are more likely to be close to the high-demand city centre (x = 0.0). Naively allocat￾ing passengers to drivers in this setting would proliferate the gender pay gap. Instead, we constrain the allocation plan T such that the subgroups have equal earning oppor￾tunity. all other variables remain unchanged and c2 j(k +1) =  vj(k) ∑… view at source ↗
Figure 3
Figure 3. Figure 3: Realised flexibility 1 v˜j [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

We consider the problem of optimally distributing resources from a set of suppliers to a set of consumers in the presence of general transportation constraints, and including heterogeneous flexibility in the marginal constraints. Such problems frequently arise in a variety of practical settings; for example, in the context of sharing economy applications, where one is not only interested in the transportation plan, but also its realisation, and in other problems that involve the study of martingales. Our principal contribution in this paper is to consider a generalisation of the classic entropically regularised Optimal Transport formulation in which such problems can be solved with a Sinkhorn algorithm. In particular, we present provably convergent Sinkhorn-like algorithms for solving this class of problems, and provide examples to both illustrate the utility of our approach as well as its efficacy.

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.

Referee Report

0 major / 3 minor

Summary. The paper generalizes the entropically regularized optimal transport problem to incorporate general transportation constraints together with heterogeneous martingale-type marginal constraints. It claims to derive Sinkhorn-like iterative scaling algorithms for this class of problems and asserts that these algorithms are provably convergent, with numerical examples illustrating their use in sharing-economy and martingale settings.

Significance. If the claimed convergence guarantees are rigorously established, the work extends the practical reach of efficient Sinkhorn-type methods to a broader family of constrained transport problems that arise in applications with flexible marginals. The explicit construction of generalized scaling procedures and the provision of examples constitute the main strengths.

minor comments (3)
  1. [Abstract] The abstract states that the algorithms are 'provably convergent' but does not list the precise assumptions (e.g., positivity or boundedness conditions on the cost and constraint matrices) under which the guarantees hold; a short statement of the main hypotheses should appear in the introduction or abstract.
  2. Notation for the generalized marginal constraints and the heterogeneous flexibility parameters is introduced without a consolidated table or diagram; adding such a reference would improve readability when the algorithms are later defined.
  3. The examples section would benefit from a brief comparison of iteration counts or run times against a standard Sinkhorn baseline on the same instances to quantify the added cost of the generalized constraints.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending minor revision. The referee's description accurately reflects the paper's focus on generalizing entropically regularized optimal transport with heterogeneous martingale-type constraints and deriving provably convergent Sinkhorn-like algorithms. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper extends the standard entropically regularized OT formulation and Sinkhorn-Knopp scaling procedure to include generalized transportation constraints and heterogeneous martingale-type marginals, presenting iterative algorithms with convergence guarantees. The abstract and description indicate a direct algorithmic generalization without any quoted self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claim to prior author work. The derivation chain remains self-contained against external OT benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract; no explicit free parameters, axioms, or invented entities are described. The existence of convergent iterative scaling for the generalized constraints is implicitly assumed but not detailed.

pith-pipeline@v0.9.1-grok · 5661 in / 1003 out tokens · 37145 ms · 2026-06-26T03:26:53.345094+00:00 · methodology

discussion (0)

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Reference graph

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