Quantitative Stability of Many-Marginal Schrodinger Bridge

Geoffrey Schiebinger; Rentian Yao; Young-Heon Kim

arxiv: 2604.15191 · v1 · submitted 2026-04-16 · 🧮 math.PR · cs.IT· math.IT

Quantitative Stability of Many-Marginal Schrodinger Bridge

Rentian Yao , Young-Heon Kim , Geoffrey Schiebinger This is my paper

Pith reviewed 2026-05-10 09:45 UTC · model grok-4.3

classification 🧮 math.PR cs.ITmath.IT

keywords Schrödinger bridgemulti-marginalquantitative stabilityKL divergenceasymptotic expansionentropic optimal transportWasserstein geodesicpath space measures

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The pith

The KL divergence between two multi-marginal Schrödinger bridges is asymptotically bounded by the terminal marginal KL plus a time-integrated squared discrepancy of Wasserstein-2 velocities and log-density gradients, with the bound tight in

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

This paper proves a quantitative stability result for Schrödinger bridges when the number of marginal constraints grows large. It establishes that the KL divergence between two such bridges on path space is controlled asymptotically by the KL divergence at the terminal marginal and an integrated term measuring squared differences between Wasserstein-2 geodesic velocities and log-density gradients. The resulting error estimates remain valid independently of the number of marginals and become tight when the marginal constraints are left unperturbed. The central technical step is an asymptotic expansion of the Schrödinger potentials to order at least two as the regularization coefficient vanishes, which also yields expansions for entropic Brenier maps and the entropic optimal transport cost under close marginals.

Core claim

When the number of marginal constraints increases, the Kullback-Leibler divergence between two multi-marginal Schrödinger bridges, viewed as measures on path space, can be asymptotically bounded by the terminal marginal KL divergence and a time-integrated squared discrepancy that combines Wasserstein-2 geodesic velocity fields with a log-density gradient term. The upper bound is asymptotically tight: it converges to zero as the number of marginal constraints increases under unperturbed marginals. The key step is an asymptotic expansion of order k ≥ 2 of the Schrödinger potentials with respect to a diminishing regularization coefficient. Byproducts include asymptotic expansions of entropic Br

What carries the argument

The asymptotic expansion of order k ≥ 2 of the Schrödinger potentials with respect to the diminishing regularization coefficient, which converts the many-marginal stability question into a controllable limit.

If this is right

Error estimates for approximations of multi-marginal problems remain valid no matter how many marginal constraints are added.
Asymptotic expansions of entropic Brenier maps become available for entropic optimal self-transport problems.
The entropic optimal transport cost admits an asymptotic expansion in the regularization coefficient when two marginal constraints are close.
The Schrödinger functional satisfies a stability property under small changes to the marginal constraints.

Where Pith is reading between the lines

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

The number-independent bound suggests that the stability statement survives passage to a continuous-time limit in which marginals are imposed at every instant.
The same expansion technique supplies higher-order corrections for entropic transport costs when marginals differ by a small but fixed amount.
Stability of the Schrödinger functional allows first-order sensitivity analysis of the bridge measure with respect to perturbations of any single marginal.

Load-bearing premise

The marginal constraints are unperturbed or sufficiently close, and the Schrödinger potentials admit an asymptotic expansion of order at least 2 as the regularization coefficient goes to zero.

What would settle it

Compute two multi-marginal Schrödinger bridges with identical marginals at every step, increase the number of time points, and check whether their path-space KL divergence converges to zero at the rate predicted by the integrated discrepancy term.

Figures

Figures reproduced from arXiv: 2604.15191 by Geoffrey Schiebinger, Rentian Yao, Young-Heon Kim.

**Figure 1.** Figure 1: Why existing stability analyses for multi-marginal Schr¨odinger bridges fail as the EOT regularization coefficient ε = m−1 → 0 +: Existing results bound the difference of Schr¨odinger potentials along two curves ρ µ t and ρ ν t using the distances between their marginal constraints ρ µ tj and ρ ν tj , which are typically of order O(1) with a poor scaling O(Poly(ε)e 1/ε) with respect to ε. As ε = 1/m → 0 +,… view at source ↗

read the original abstract

In this paper, we explore quantitative stability of multi-marginal Schr\"odinger bridges with respect to the marginal constraints. We focus on the case where the number of marginal constraints is large (i.e. ``many-marginals"). When this number increases, we show that the Kullback--Leibler (KL) divergence between two multi-marginal Schr\"odinger bridges, as measures on the path space, can be asymptotically bounded by the terminal marginal KL divergence and a time-integrated squared discrepancy {that combines} Wasserstein-2 geodesic velocity fields with a log-density gradient term. Our stability upper bound is also asymptotically tight: it converges to zero as the number of marginal constraints increases with unperturbed marginal constraints. To the best of our knowledge, this is the first such stability result that addresses the many-marginal regime, giving error estimates that are asymptotically independent of the number of marginals. To achieve our result, the key step is to derive an asymptotic expansion (of order $k\ge 2$) of Schr\"odinger potentials with respect to a diminishing regularization coefficient. This result can also be applied to deriving asymptotic expansions of entropic Brenier maps in entropic optimal self-transport problems. As byproducts of our analyses, we also establish the asymptotic expansion of entropic optimal transport cost with respect to the diminishing regularization coefficient when two marginal constraints are sufficiently close. We also prove a stability property of the Schr\"odinger functional.

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

This paper gives the first quantitative stability bounds for many-marginal Schrödinger bridges that stay controlled as the number of marginals grows, via an asymptotic expansion of the potentials.

read the letter

The main point is that this work establishes the first stability result for Schrödinger bridges in the many-marginal regime, with KL bounds between bridges that are asymptotically independent of the number of marginal constraints. The authors derive an order-k≥2 expansion of the Schrödinger potentials as the regularization parameter goes to zero, then use it to control the path-space KL divergence by the terminal marginal KL plus a time-integrated discrepancy that mixes Wasserstein-2 geodesic velocities with a log-density gradient term. They also show the bound is asymptotically tight under unperturbed marginals and obtain byproducts on the expansion of entropic OT cost for close marginals plus stability of the Schrödinger functional. These steps extend earlier two-marginal work in a direct way and avoid obvious circularity by starting from standard bridge theory.

Referee Report

1 major / 0 minor

Summary. The manuscript establishes quantitative stability for multi-marginal Schrödinger bridges in the many-marginal regime. It shows that the KL divergence between two such bridges (as path-space measures) is asymptotically bounded above by the terminal marginal KL divergence plus a time-integrated squared discrepancy combining Wasserstein-2 geodesic velocity fields and a log-density gradient term. The upper bound is asymptotically tight and converges to zero as the number of marginal constraints N increases under unperturbed marginals, yielding error estimates that are asymptotically independent of N. The key technical step is an asymptotic expansion of order k ≥ 2 for the Schrödinger potentials as the regularization coefficient ε → 0; byproducts include expansions for entropic Brenier maps, the entropic OT cost under close marginals, and a stability property of the Schrödinger functional.

Significance. If the central claims hold, the work supplies the first stability estimates for Schrödinger bridges that remain valid and asymptotically independent of the number of marginal constraints. This is potentially significant for applications involving high-dimensional or multi-marginal entropic optimal transport, where controlling dependence on N is essential. The byproduct asymptotic expansions for entropic OT costs and Brenier maps are also useful. The significance is tempered by the need to confirm uniformity of the potential expansion with respect to N.

major comments (1)

[Key technical step (asymptotic expansion of Schrödinger potentials)] The claimed asymptotic independence of the stability bound from N rests on the order-k≥2 asymptotic expansion of the Schrödinger potentials (the key technical step identified in the abstract). The derivation must be shown to be uniform in N: if the expansion coefficients or remainder terms grow with N, the time-integrated discrepancy term would acquire hidden N-dependent factors, violating the central claim of N-independent error estimates. Please supply explicit N-uniform bounds or estimates on the expansion in the relevant theorem or section establishing the expansion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need to explicitly confirm N-uniformity in the asymptotic expansion of the Schrödinger potentials. This is a helpful observation that will improve the clarity of our presentation. We address the major comment point by point below.

read point-by-point responses

Referee: [Key technical step (asymptotic expansion of Schrödinger potentials)] The claimed asymptotic independence of the stability bound from N rests on the order-k≥2 asymptotic expansion of the Schrödinger potentials (the key technical step identified in the abstract). The derivation must be shown to be uniform in N: if the expansion coefficients or remainder terms grow with N, the time-integrated discrepancy term would acquire hidden N-dependent factors, violating the central claim of N-independent error estimates. Please supply explicit N-uniform bounds or estimates on the expansion in the relevant theorem or section establishing the expansion.

Authors: We thank the referee for this precise comment. The asymptotic expansion of the Schrödinger potentials is established in Theorem 3.1 via a perturbative fixed-point argument around the unregularized multi-marginal OT problem. The contraction mapping constant and the bounds on the remainder (of order k ≥ 2) are derived from the uniform Lipschitz and boundedness assumptions on the log-densities of the marginals together with the fixed reference measure (Brownian motion); these quantities are independent of N by the problem setup in the many-marginal regime. Consequently, the constants appearing in the time-integrated discrepancy term of the stability bound remain free of N. To make this uniformity fully explicit, we will add a dedicated lemma (new Lemma 3.2) that isolates the N-independent estimates on the expansion coefficients and remainder. This addition will be included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; central stability bound rests on independently derived asymptotic expansion of potentials

full rationale

The paper identifies its key technical step as deriving (not assuming or fitting) an order-k≥2 asymptotic expansion of Schrödinger potentials as the regularization coefficient vanishes. This expansion is then used to control the KL divergence bound between many-marginal bridges. No load-bearing step reduces by construction to the target stability statement, to a fitted parameter, or to a self-citation chain; the abstract and description present the expansion as a new derivation from standard multi-marginal Schrödinger bridge theory. The claimed asymptotic independence from the number of marginals follows from controlling the expansion uniformly under the stated closeness assumptions on the marginals, without tautological re-use of the final bound. This is the normal case of a self-contained analytic derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on standard assumptions in Schrödinger bridge and optimal transport theory while introducing a new asymptotic expansion technique as its main contribution. No free parameters or new entities are mentioned in the abstract.

axioms (2)

domain assumption The multi-marginal Schrödinger bridge exists and is unique under the given marginal constraints.
Implicit in the setup of the problem as standard in the field.
ad hoc to paper Schrödinger potentials admit an asymptotic expansion of order at least 2 as the regularization coefficient diminishes.
This is presented as the key step derived in the paper to achieve the stability result.

pith-pipeline@v0.9.0 · 5564 in / 1410 out tokens · 34819 ms · 2026-05-10T09:45:56.548349+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

+ 2 ∇v† 0,∇(u ku† 0) +u ku† 0∆v† 0 = 1 2∆(uku†

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[2]

Therefore, we have vkA0 +v 0Ak = 1 2∆(uku†

+ ∇logv † 0,∇(u ku† 0) + v0∆v† 0 2 ·u ku† 0. Therefore, we have vkA0 +v 0Ak = 1 2∆(uku†

work page
[3]

+ ∇logv † 0,∇(u ku† 0) + h v0∆v† 0 2 +S 0 i uku† 0 + v0 k−1X l=0 ∆k−l+1(ulρ0) 2k−l+1(k−l+ 1)! −S 0Sk−1 Now, let us calculate the second term in (D.1). Note that we have u0Bk +u kB0 =u 0 kX l=0 lX i=0 ∆k+1−l(viρl−i) 2k+1−l(k+ 1−l)! + kX i=0 viρk−i+1 +u kB0 = h ukB0 + u0∆(vkρ0) 2 +u 0vkρ1 i +u 0 k−1X l=0 lX i=0 ∆k+1−l(v0ρl−i) 2k+1−l(k+ 1−l)! + k−1X i=0 ∆(vi...

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[4]

= u0 2 ∆ u† 0(Sk−1 −u † 0uk) =− u0 2 ∆[u† 0(u† 0uk)] + u0 2 ∆(u† 0Sk−1) =− u0 2 u† 0∆(u† 0uk) + 2 ∇u† 0,∇(u † 0uk) +u † 0uk∆u† 0 + u0 2 ∆(u† 0Sk−1) =− 1 2∆(u† 0uk)− ∇logu † 0,∇(u † 0uk) − u0∆u† 0 2 u† 0uk + u0 2 ∆(u† 0Sk−1), and we also have u0vkρ1 =u 0ρ1v0v† 0vk =u 0ρ1v0(Sk−1 −u † 0uk) =−u 0v0ρ1u† 0uk +u 0v0ρ1Sk−1. Therefore, we have u0Bk +u kB0 =− 1 2∆(...

work page
[5]

+ ∇logρ 0,∇(u † 0uk) + h v0∆v† 0 2 +S 0 +u 0v0ρ1 + u0∆u† 0 2 −B 0u0 i u† 0uk −u 0 ∆(u† 0Sk−1) 2 +v 0ρ1Sk−1 + k−1X l=0 lX i=0 ∆k+1−l(v0ρl−i) 2k+1−l(k+ 1−l)! + k−1X i=0 ∆(viρk−i) 2 + k−1X i=0 viρk−i+1 + v0 k−1X l=0 ∆k−l+1(ulρ0) 2k−l+1(k−l+ 1)! −S 0Sk−1 . Note that we have v0∆v† 0 2 +S 0 +u 0v0ρ1 + u0∆u† 0 2 −B 0u0 = u0v0 2 v† 0∆u† 0 −u † 0∆v† 0 + 2ρ1 + 2S0 ...

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[6]

+ ∇logρ 0,∇(u † 0uk) = 1 ρ0 ∇ · ρ0(u† 0uk) . Therefore, we have vkA0 +v 0Ak − ukB0 +u 0Bk = 1 ρ0 ∇ · ρ0∇(u† 0uk) + v0 k−1X l=0 ∆k−l+1(ulρ0) 2k−l+1(k−l+ 1)! −S 0Sk−1 −u 0 ∆(u† 0Sk−1) 2 +v 0ρ1Sk−1 + k−1X l=0 lX i=0 ∆k+1−l(v0ρl−i) 2k+1−l(k+ 1−l)! + k−1X i=0 ∆(viρk−i) 2 + k−1X i=0 viρk−i+1 . Combining with (D.1) yields ∇ · ρ0∇(u† 0uk) =ρ 0 k−1X i=1 uk−iBi − k...

work page

[1] [1]

+ 2 ∇v† 0,∇(u ku† 0) +u ku† 0∆v† 0 = 1 2∆(uku†

work page

[2] [2]

Therefore, we have vkA0 +v 0Ak = 1 2∆(uku†

+ ∇logv † 0,∇(u ku† 0) + v0∆v† 0 2 ·u ku† 0. Therefore, we have vkA0 +v 0Ak = 1 2∆(uku†

work page

[3] [3]

+ ∇logv † 0,∇(u ku† 0) + h v0∆v† 0 2 +S 0 i uku† 0 + v0 k−1X l=0 ∆k−l+1(ulρ0) 2k−l+1(k−l+ 1)! −S 0Sk−1 Now, let us calculate the second term in (D.1). Note that we have u0Bk +u kB0 =u 0 kX l=0 lX i=0 ∆k+1−l(viρl−i) 2k+1−l(k+ 1−l)! + kX i=0 viρk−i+1 +u kB0 = h ukB0 + u0∆(vkρ0) 2 +u 0vkρ1 i +u 0 k−1X l=0 lX i=0 ∆k+1−l(v0ρl−i) 2k+1−l(k+ 1−l)! + k−1X i=0 ∆(vi...

work page

[4] [4]

= u0 2 ∆ u† 0(Sk−1 −u † 0uk) =− u0 2 ∆[u† 0(u† 0uk)] + u0 2 ∆(u† 0Sk−1) =− u0 2 u† 0∆(u† 0uk) + 2 ∇u† 0,∇(u † 0uk) +u † 0uk∆u† 0 + u0 2 ∆(u† 0Sk−1) =− 1 2∆(u† 0uk)− ∇logu † 0,∇(u † 0uk) − u0∆u† 0 2 u† 0uk + u0 2 ∆(u† 0Sk−1), and we also have u0vkρ1 =u 0ρ1v0v† 0vk =u 0ρ1v0(Sk−1 −u † 0uk) =−u 0v0ρ1u† 0uk +u 0v0ρ1Sk−1. Therefore, we have u0Bk +u kB0 =− 1 2∆(...

work page

[5] [5]

+ ∇logρ 0,∇(u † 0uk) + h v0∆v† 0 2 +S 0 +u 0v0ρ1 + u0∆u† 0 2 −B 0u0 i u† 0uk −u 0 ∆(u† 0Sk−1) 2 +v 0ρ1Sk−1 + k−1X l=0 lX i=0 ∆k+1−l(v0ρl−i) 2k+1−l(k+ 1−l)! + k−1X i=0 ∆(viρk−i) 2 + k−1X i=0 viρk−i+1 + v0 k−1X l=0 ∆k−l+1(ulρ0) 2k−l+1(k−l+ 1)! −S 0Sk−1 . Note that we have v0∆v† 0 2 +S 0 +u 0v0ρ1 + u0∆u† 0 2 −B 0u0 = u0v0 2 v† 0∆u† 0 −u † 0∆v† 0 + 2ρ1 + 2S0 ...

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[6] [6]

+ ∇logρ 0,∇(u † 0uk) = 1 ρ0 ∇ · ρ0(u† 0uk) . Therefore, we have vkA0 +v 0Ak − ukB0 +u 0Bk = 1 ρ0 ∇ · ρ0∇(u† 0uk) + v0 k−1X l=0 ∆k−l+1(ulρ0) 2k−l+1(k−l+ 1)! −S 0Sk−1 −u 0 ∆(u† 0Sk−1) 2 +v 0ρ1Sk−1 + k−1X l=0 lX i=0 ∆k+1−l(v0ρl−i) 2k+1−l(k+ 1−l)! + k−1X i=0 ∆(viρk−i) 2 + k−1X i=0 viρk−i+1 . Combining with (D.1) yields ∇ · ρ0∇(u† 0uk) =ρ 0 k−1X i=1 uk−iBi − k...

work page