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arxiv: 2510.03803 · v2 · submitted 2025-10-04 · 🧮 math.OC · cs.NA· math.NA

Well-Posedness and Efficient Algorithms for Inverse Optimal Transport with Bregman Regularization

Chenglong Bao , Zanyu Li , Yunan Yang This is my paper

Pith reviewed 2026-05-18 10:33 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords inverse optimal transportBregman regularizationwell-posednessblock coordinate descentstabilityexistence and uniquenessconvergence rateoptimal transport
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The pith

The inverse optimal transport problem with Bregman regularization is well-posed under cost matrix assumptions and solved efficiently by block coordinate descent.

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

This paper proves that the inverse optimal transport problem regularized by Bregman divergence has solutions that exist, are unique up to equivalence, and remain stable when the cost matrix obeys certain structural conditions. It also introduces an inexact block coordinate descent algorithm that converges linearly and becomes especially fast for quadratic penalties because the Hessian is diagonal, allowing simple element-wise Newton steps. Readers interested in recovering hidden costs from transport data in fields like economics or machine learning would find these guarantees and computational tools useful for reliable and scalable applications.

Core claim

Under structural assumptions on the cost matrix, the Bregman-regularized inverse optimal transport problem admits existence and uniqueness of solutions up to equivalence classes, along with stability. A sufficient condition guarantees existence under general constraints, and an inexact BCD method is developed with linear convergence, where quadratic penalties enable efficient element-wise Newton updates due to diagonal Hessians.

What carries the argument

Bregman-regularized formulation of inverse optimal transport, solved by inexact block coordinate descent with diagonal Hessian exploitation for quadratic cases.

If this is right

  • The recovered cost is stable to perturbations in the observed data.
  • The algorithm converges linearly to the solution.
  • Element-wise updates make computation efficient for quadratic penalties.
  • Existence holds under a provided sufficient condition on constraints.
  • Numerical tests confirm the theory on datasets including marriage matching.

Where Pith is reading between the lines

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

  • The stability property could improve robustness in applications with measurement noise.
  • Similar descent methods might apply to other regularized inverse problems in optimization.
  • Extensions to continuous settings or different divergences could follow from the discrete analysis.
  • Practical implementations may benefit from the fast updates in high-dimensional matching tasks.

Load-bearing premise

The cost matrix must satisfy specific structural assumptions to ensure existence, uniqueness, and stability of the inverse problem solutions.

What would settle it

A numerical example or counterexample demonstrating non-uniqueness or instability for a cost matrix that fails the structural assumptions would disprove the well-posedness results.

Figures

Figures reproduced from arXiv: 2510.03803 by Chenglong Bao, Yunan Yang, Zanyu Li.

Figure 1
Figure 1. Figure 1: Experiment results of verification of stability bounds for fo [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Experiment results of effect of regularization parameter [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Experiment results of effect of regularization parameter [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
read the original abstract

This work analyzes the inverse optimal transport (IOT) problem under Bregman regularization. We establish well-posedness results, including existence, uniqueness (up to equivalence classes of solutions), and stability, under several structural assumptions on the cost matrix. On the computational side, we investigate the existence of solutions to the optimization problem with general constraints on the cost matrix and provide a sufficient condition guaranteeing existence. In addition, we propose an inexact block coordinate descent (BCD) method for the problem with a strongly convex penalty term. In particular, when the penalty is quadratic, the subproblems admit a diagonal Hessian structure, which enables highly efficient element-wise Newton updates. We establish a linear convergence rate for the algorithm and demonstrate its practical performance through numerical experiments, including the validation of stability bounds, the investigation of regularization effects, and the application to a marriage matching dataset.

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

2 major / 2 minor

Summary. This paper analyzes the inverse optimal transport (IOT) problem under Bregman regularization. It establishes well-posedness results including existence, uniqueness up to equivalence classes of solutions, and stability under several structural assumptions on the cost matrix. It provides a sufficient condition for existence under general constraints on the cost matrix and proposes an inexact block coordinate descent (BCD) method for strongly convex penalties. For quadratic penalties, subproblems admit a diagonal Hessian structure enabling efficient element-wise Newton updates, with a proven linear convergence rate. Numerical experiments validate stability bounds, investigate regularization effects, and apply the method to a marriage matching dataset.

Significance. If the well-posedness and convergence results hold under the stated assumptions, the work provides a useful theoretical and algorithmic contribution to inverse problems in optimal transport, with potential applications in matching problems and data-driven inference. The combination of stability analysis, existence conditions, and an efficient linearly convergent algorithm with explicit Newton updates strengthens the practical utility. The numerical validation on both synthetic stability checks and a real dataset adds supporting evidence for the claims.

major comments (2)
  1. [§3] §3 (well-posedness): the structural assumptions on the cost matrix invoked for uniqueness up to equivalence classes and stability are load-bearing for the central claim; they should be stated explicitly with a discussion of whether they are necessary or can be weakened, including a concrete counterexample if the assumptions are violated.
  2. [Theorem 5.3] Theorem 5.3 (linear convergence): the proof of the linear rate for the inexact BCD method under quadratic penalty relies on the diagonal-Hessian Newton updates; verify that the inexactness tolerance does not degrade the rate constant in a manner dependent on the Bregman regularization parameter.
minor comments (2)
  1. [Abstract] The abstract refers to 'several structural assumptions' without naming them; a one-sentence summary of the key assumptions would improve readability.
  2. [Numerical Experiments] In the numerical section, the captions for figures showing stability bounds should explicitly list the matrix dimensions, regularization values, and number of trials used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [§3] §3 (well-posedness): the structural assumptions on the cost matrix invoked for uniqueness up to equivalence classes and stability are load-bearing for the central claim; they should be stated explicitly with a discussion of whether they are necessary or can be weakened, including a concrete counterexample if the assumptions are violated.

    Authors: We agree that the structural assumptions on the cost matrix are central to the uniqueness-up-to-equivalence and stability results in Section 3. In the revised manuscript we will restate these assumptions explicitly at the opening of the section. We will add a dedicated paragraph discussing their necessity, the extent to which they can be relaxed, and a concrete counter-example (a 2-by-2 cost matrix violating the assumption) that produces multiple non-equivalent solutions. These additions will be incorporated in the next version. revision: yes

  2. Referee: [Theorem 5.3] Theorem 5.3 (linear convergence): the proof of the linear rate for the inexact BCD method under quadratic penalty relies on the diagonal-Hessian Newton updates; verify that the inexactness tolerance does not degrade the rate constant in a manner dependent on the Bregman regularization parameter.

    Authors: We have re-examined the proof of Theorem 5.3. The linear rate is obtained by showing that the inexact block updates produce a contraction whose constant depends only on the strong-convexity modulus of the quadratic penalty and on the step-size parameters; the tolerance threshold can be chosen independently of the Bregman regularization parameter λ. The diagonal Hessian structure further ensures that the local Newton error remains uniformly controlled. We will insert a short clarifying remark after the theorem statement making this independence explicit. No alteration of the theorem itself is required. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results on well-posedness (existence, uniqueness up to equivalence classes, and stability) for inverse optimal transport under Bregman regularization are derived from standard convex analysis and optimal transport theory under explicit structural assumptions on the cost matrix. The algorithmic contribution—an inexact block coordinate descent method with linear convergence for quadratic penalties—relies on diagonal Hessian structure and standard convergence arguments for block coordinate methods, independent of any data-fitting step or self-referential definition. No step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the derivation chain is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on structural assumptions about the cost matrix and standard properties of Bregman divergences and strongly convex penalties; no free parameters or new invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Structural assumptions on the cost matrix
    Invoked to guarantee existence, uniqueness up to equivalence classes, and stability of the inverse problem.

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    Sparsist ency for inverse optimal transport

    Francisco Andrade, Gabriel Peyr´ e, and Clarice Poon. Sparsist ency for inverse optimal transport. arXiv preprint arXiv:2310.05461, 2023

    work page arXiv 2023

  2. [2]

    Learning from samples: Inverse problems over measures via sharpened fenchel-young losses

    Francisco Andrade, Gabriel Peyr´ e, and Clarice Poon. Learning from samples: Inverse problems over measures via sharpened fenchel-young losses. arXiv preprint arXiv:2505.07124 , 2025

    work page arXiv 2025

  3. [3]

    Sis ta: learning optimal transport costs under sparsity constraints

    Guillaume Carlier, Arnaud Dupuy, Alfred Galichon, and Yifei Sun. Sis ta: learning optimal transport costs under sparsity constraints. Communications on Pure and Applied Mathematics , 76(9):1659–1677, 2023

    work page 2023

  4. [4]

    Discrete probabilist ic inverse optimal transport

    Wei-Ting Chiu, Pei Wang, and Patrick Shafto. Discrete probabilist ic inverse optimal transport. In International Conference on Machine Learning , pages 3925–3946. PMLR, 2022

    work page 2022

  5. [5]

    Optimal transport for domain adaptation

    Nicolas Courty, R´ emi Flamary, Devis Tuia, and Alain Rakotomamon jy. Optimal transport for domain adaptation. IEEE transactions on pattern analysis and machine intellig ence, 39(9):1853–1865, 2016

    work page 2016

  6. [6]

    Performa nce of recommender algorithms on top-n recommendation tasks

    Paolo Cremonesi, Yehuda Koren, and Roberto Turrin. Performa nce of recommender algorithms on top-n recommendation tasks. In Proceedings of the fourth ACM conference on Recommender sys tems, pages 39–46, 2010

    work page 2010

  7. [7]

    Sinkhorn distances: Lightspeed computation of o ptimal transport

    Marco Cuturi. Sinkhorn distances: Lightspeed computation of o ptimal transport. Advances in neural information processing systems , 26, 2013

    work page 2013

  8. [8]

    Regula rized optimal transport and the rot mover’s distance

    Arnaud Dessein, Nicolas Papadakis, and Jean-Luc Rouas. Regula rized optimal transport and the rot mover’s distance. Journal of Machine Learning Research , 19(15):1–53, 2018

    work page 2018

  9. [9]

    Estimating matchin g affinity matrices under low-rank constraints

    Arnaud Dupuy, Alfred Galichon, and Yifei Sun. Estimating matchin g affinity matrices under low-rank constraints. Information and Inference: A Journal of the IMA , 8(4):677–689, 2019

    work page 2019

  10. [10]

    Quadratically regularized op timal transport on graphs

    Montacer Essid and Justin Solomon. Quadratically regularized op timal transport on graphs. SIAM Journal on Scientific Computing , 40(4):A1961–A1986, 2018

    work page 2018

  11. [11]

    Entropy-regularized optimal transport for machine learning

    Aude Genevay. Entropy-regularized optimal transport for machine learning. PhD thesis, Universit´ e Paris sciences et lettres, 2019

    work page 2019

  12. [12]

    Iden tifiability of the optimal transport cost on finite spaces

    Alberto Gonz´ alez-Sanz, Michel Groppe, and Axel Munk. Iden tifiability of the optimal transport cost on finite spaces. arXiv preprint arXiv:2410.23146 , 2024

    work page arXiv 2024

  13. [13]

    Nonlin ear inverse optimal transport: Identi- fiability of the transport cost from its marginals and optimal values

    Alberto Gonz´ alez-Sanz, Michel Groppe, and Axel Munk. Nonlin ear inverse optimal transport: Identi- fiability of the transport cost from its marginals and optimal values. SIAM Journal on Mathematical Analysis, 56(6):7808–7829, 2024

    work page 2024

  14. [14]

    Optimal mass transport: Signal processing and machine-learning applications

    Soheil Kolouri, Se Rim Park, Matthew Thorpe, Dejan Slepcev, an d Gustavo K Rohde. Optimal mass transport: Signal processing and machine-learning applications. IEEE signal processing magazine , 34(4):43–59, 2017

    work page 2017

  15. [15]

    Matrix factoriza tion techniques for recommender systems

    Yehuda Koren, Robert Bell, and Chris Volinsky. Matrix factoriza tion techniques for recommender systems. Computer, 42(8):30–37, 2009

    work page 2009

  16. [16]

    Learning to match via inverse optimal transport

    Ruilin Li, Xiaojing Ye, Haomin Zhou, and Hongyuan Zha. Learning to match via inverse optimal transport. Journal of machine learning research , 20(80):1–37, 2019

    work page 2019

  17. [17]

    Learning tra nsport cost from subset correspon- dence

    Ruishan Liu, Akshay Balsubramani, and James Zou. Learning tra nsport cost from subset correspon- dence. arXiv preprint arXiv:1909.13203 , 2019. 26

    work page arXiv 1909

  18. [18]

    Quadratically r egularized optimal transport

    Dirk A Lorenz, Paul Manns, and Christian Meyer. Quadratically r egularized optimal transport. Applied Mathematics & Optimization , 83(3):1919–1949, 2021

    work page 1919

  19. [19]

    Error bounds and convergence analysis of feasible descent methods: a general approach

    Zhi-Quan Luo and Paul Tseng. Error bounds and convergence analysis of feasible descent methods: a general approach. Annals of Operations Research , 46(1):157–178, 1993

    work page 1993

  20. [20]

    Learning cost functions for optimal transport

    Shaojun Ma, Haodong Sun, Xiaojing Ye, Hongyuan Zha, and Hao min Zhou. Learning cost functions for optimal transport. arXiv preprint arXiv:2002.09650 , 2020

    work page arXiv 2002

  21. [21]

    Probabilistic matrix factor ization

    Andriy Mnih and Russ R Salakhutdinov. Probabilistic matrix factor ization. Advances in neural infor- mation processing systems , 20, 2007

    work page 2007

  22. [22]

    arXiv preprint arXiv:2410.02628 , year=

    Mikhail Persiianov, Arip Asadulaev, Nikita Andreev, Nikita Starod ubcev, Dmitry Baranchuk, Anastasis Kratsios, Evgeny Burnaev, and Alexander Korotin. Inverse entr opic optimal transport solves semi- supervised learning via data likelihood maximization. arXiv preprint arXiv:2410.02628 , 2024

    work page arXiv 2024

  23. [23]

    Computational optimal transport: With applications to data science

    Gabriel Peyr´ e, Marco Cuturi, et al. Computational optimal transport: With applications to data science. Foundations and Trends ® in Machine Learning , 11(5-6):355–607, 2019

    work page 2019

  24. [24]

    A semismooth newton method for the nearest euclid ean distance matrix problem

    Hou-Duo Qi. A semismooth newton method for the nearest euclid ean distance matrix problem. SIAM Journal on Matrix analysis and applications , 34(1):67–93, 2013

    work page 2013

  25. [25]

    Convex analysis , volume 28

    R Tyrrell Rockafellar. Convex analysis , volume 28. Princeton university press, 1997

    work page 1997

  26. [26]

    Understanding and generalizing contrastive learning from the inverse optimal transport perspec tive

    Liangliang Shi, Gu Zhang, Haoyu Zhen, Jintao Fan, and Junchi Ya n. Understanding and generalizing contrastive learning from the inverse optimal transport perspec tive. In International conference on machine learning , pages 31408–31421. PMLR, 2023

    work page 2023

  27. [27]

    Inverse optimal transport

    Andrew M Stuart and Marie-Therese Wolfram. Inverse optimal transport. SIAM Journal on Applied Mathematics, 80(1):599–619, 2020

    work page 2020

  28. [28]

    Optimal transport: old and new , volume 338

    C´ edric Villani et al. Optimal transport: old and new , volume 338. Springer, 2008

    work page 2008

  29. [29]

    Self-supervised vide o summarization guided by semantic inverse optimal transport

    Yutong Wang, Hongteng Xu, and Dixin Luo. Self-supervised vide o summarization guided by semantic inverse optimal transport. In Proceedings of the 31st ACM International Conference on Mul timedia, pages 6611–6622, 2023

    work page 2023

  30. [30]

    Explainable legal case matching via inverse optimal transport-base d rationale extraction

    Weijie Yu, Zhongxiang Sun, Jun Xu, Zhenhua Dong, Xu Chen, Hon gteng Xu, and Ji-Rong Wen. Explainable legal case matching via inverse optimal transport-base d rationale extraction. In Proceedings of the 45th international ACM SIGIR conference on research a nd development in information retrieval , pages 657–668, 2022. 27

    work page 2022