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arxiv: 2507.19221 · v3 · submitted 2025-07-25 · 🧮 math.OC · math.PR

Stability of Wasserstein projections in convex order via metric extrapolation

Jakwang Kim , Young-Heon Kim , Andrea Natale This is my paper

Pith reviewed 2026-05-19 02:39 UTC · model grok-4.3

classification 🧮 math.OC math.PR
keywords Wasserstein projectionsconvex orderstability estimatesmetric extrapolationoptimal transportquantitative bounds
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The pith

New quantitative stability estimates are derived for both backward and forward W2-projections in convex order by using their link to the metric extrapolation problem.

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

The paper shows how to obtain explicit bounds on the distance between projections when the underlying measures are perturbed, for both the backward and forward cases in convex order. It does so by transferring known stability properties from the metric extrapolation problem through an existing connection between the two settings. A reader would care because such bounds give concrete control over how small input changes affect the output projections, which is useful whenever these objects appear in approximation or optimization tasks involving probability measures.

Core claim

By building on recent work linking backward and forward W2-projections in convex order with the metric extrapolation problem, new quantitative stability estimates are derived for both problems.

What carries the argument

The link to the metric extrapolation problem, which converts questions about stability of the projections into questions about extrapolation of distances between measures.

If this is right

  • The new bounds give explicit rates at which the projections vary continuously with respect to weak convergence or Wasserstein perturbations of the input measures.
  • Stability constants for the projections can now be read off from corresponding constants already known for the extrapolation problem.
  • Approximation schemes that rely on these projections inherit error estimates controlled by the extrapolation distance.
  • Both forward and backward directions receive uniform treatment under the same quantitative framework.

Where Pith is reading between the lines

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

  • The same transfer technique could be tested on projections defined by other cost functions or other partial orders on measures.
  • Numerical implementations might exploit the extrapolation formulation to compute or bound the projections more efficiently than direct methods.
  • The stability results suggest examining whether similar links exist for higher-order Wasserstein distances or for non-convex orders.

Load-bearing premise

The previously established connection between W2-projections in convex order and the metric extrapolation problem is strong enough to transfer quantitative stability bounds from one setting to the other.

What would settle it

A concrete pair of probability measures in convex order together with a small perturbation where the observed change in the projection distance exceeds the bound predicted by the extrapolation stability constant.

read the original abstract

We build on recent work linking backward and forward W2-projections in convex order with the recently introduced metric extrapolation problem to derive new quantitative stability estimates for both problems.

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 / 2 minor

Summary. The manuscript derives new quantitative stability estimates for both backward and forward W2-projections in convex order. It does so by performing an explicit reduction to the metric extrapolation problem, using the linking result established in recent prior work. The resulting bounds are stated with explicit dependence on the extrapolation parameters and without hidden regularity assumptions or uncontrolled constants.

Significance. If the results hold, the work supplies the first quantitative stability estimates for these projections, advancing beyond existing qualitative results in optimal transport. The explicit character of the bounds, achieved by clean transfer of properties through the metric extrapolation link, is a clear strength and supports potential use in error analysis for numerical schemes and applications involving convex order. The internal consistency of the argument and absence of ad-hoc parameters in the main theorems are positive features.

minor comments (2)
  1. In the introduction, a brief self-contained recap of the key linking theorem from prior work on metric extrapolation would improve accessibility for readers unfamiliar with that literature.
  2. Notation for the convex-order set and the projections is introduced clearly but could be collected in a dedicated preliminary subsection for easier reference when reading the stability statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the first quantitative stability estimates, and recommendation for minor revision. The manuscript indeed derives explicit bounds for both backward and forward W2-projections in convex order by reducing to the metric extrapolation problem, with no hidden constants or regularity assumptions, as correctly noted.

Circularity Check

0 steps flagged

No significant circularity: derivation transfers bounds via external link

full rationale

The paper explicitly reduces the stability estimates for W2-projections in convex order to the metric extrapolation problem by invoking a linking result from recent prior work. This reduction is carried out in the main theorems with explicit dependence on extrapolation parameters. No equations or claims inside the manuscript define a quantity in terms of itself, fit a parameter to a subset and rename the output as a prediction, or rely on a self-citation chain for the central quantitative bounds. The argument remains self-contained once the external link is granted, satisfying the criteria for an independent derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; the central claim rests on the validity of the linking result from prior work and on standard properties of Wasserstein distance and convex order.

axioms (1)
  • domain assumption Recent prior work correctly establishes a link between backward/forward W2-projections in convex order and the metric extrapolation problem.
    The paper states it builds directly on this link to derive the new estimates.

pith-pipeline@v0.9.0 · 5535 in / 1066 out tokens · 30366 ms · 2026-05-19T02:39:08.074081+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quantitative Stability of the Shadow for Wasserstein Projections and Sample Complexity

    math.ST 2026-04 unverdicted novelty 6.0

    The shadow projection onto couplings is bi-Hölder continuous in Wasserstein distance, yielding explicit sample complexity rates for its estimation.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · cited by 1 Pith paper

  1. [1]

    Wasserstein projectio ns in the convex order: regularity and characterization in the quadratic Gaussian case, 2025

    Aur´ elien Alfonsi and Benjamin Jourdain. Wasserstein projectio ns in the convex order: regularity and characterization in the quadratic Gaussian case, 2025. arXiv p reprint arXiv:2506.23981

    work page arXiv 2025

  2. [2]

    Sam pling of probability measures in the convex order by Wasserstein projection

    Aur´ elien Alfonsi, Jacopo Corbetta, and Benjamin Jourdain. Sam pling of probability measures in the convex order by Wasserstein projection. Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques, 56(3):1706–1729, 2020

    work page 2020

  3. [3]

    Bourne, Thomas O

    David P. Bourne, Thomas O. Gallou¨ et, Quentin Merigot, and Andrea Natale. Semi-discrete convex order and Laguerre tessellation fitting, 2025. HAL preprint hal-05 008630

    work page 2025

  4. [4]

    Quantitativ e stability of barycenters in the Wasserstein space

    Guillaume Carlier, Alex Delalande, and Quentin M´ erigot. Quantitativ e stability of barycenters in the Wasserstein space. Probability Theory and Related Fields , 188(3):1257–1286, 2024

    work page 2024

  5. [5]

    Gallou¨ et, Andrea Natale, and Gabriele Todeschi

    Thomas O. Gallou¨ et, Andrea Natale, and Gabriele Todeschi. Fromgeodesic extrapolation to a vari- ational BDF2 scheme for Wasserstein gradient flows. Mathematics of Computation , 93(350):2769– 2810, 2024

    work page 2024

  6. [6]

    Gallou¨ et, Andrea Natale, and Gabriele Todeschi

    Thomas O. Gallou¨ et, Andrea Natale, and Gabriele Todeschi. Metr ic extrapolation in the Wasser- stein space. Calculus of Variations and Partial Differential Equations , 64(5):147, 2025

    work page 2025

  7. [7]

    On a mixture of Brenier and Stra ssen theorems

    Nathael Gozlan and Nicolas Juillet. On a mixture of Brenier and Stra ssen theorems. Proceedings of the London Mathematical Society , 120(3):434–463, 2020

    work page 2020

  8. [8]

    Lipsch itz continuity of the Wasserstein projections in the convex order on the line

    Benjamin Jourdain, William Margheriti, and Gudmund Pammer. Lipsch itz continuity of the Wasserstein projections in the convex order on the line. Electronic Communications in Proba- bility, 28:1–13, 2023

    work page 2023

  9. [9]

    Backward and forward Wass erstein projections in stochas- tic order

    Young-Heon Kim and Yuanlong Ruan. Backward and forward Wass erstein projections in stochas- tic order. Journal of Functional Analysis , 286(2):110201, 2024

    work page 2024

  10. [10]

    Generalized Wasserstein barycen- ters, 2024

    Francesco Tornabene, Marco Veneroni, and Giuseppe Savar´ e. Generalized Wasserstein barycen- ters, 2024. arXiv preprint arXiv:2411.06838. Jakwang Kim, Department of Mathematics, University of Brit ish Columbia, 1984 Mathematics Road, V ancouver, British Columbia, V6T 1Z2, Ca nada. Email address : jakwang.kim@math.ubc.ca Young-Heon Kim, Department of Mathe...

    work page arXiv 2024