Quantitative Stability of Wasserstein Barycenters over Alexandrov Spaces with Lower Curvature Bounds

Bang-Xian Han; Zhuo-Nan Zhu

arxiv: 2605.25448 · v1 · pith:DGX5PKJ5new · submitted 2026-05-25 · 🧮 math.MG · math.FA· math.PR

Quantitative Stability of Wasserstein Barycenters over Alexandrov Spaces with Lower Curvature Bounds

Bang-Xian Han , Zhuo-Nan Zhu This is my paper

Pith reviewed 2026-06-29 19:49 UTC · model grok-4.3

classification 🧮 math.MG math.FAmath.PR

keywords Wasserstein barycenterAlexandrov spacequantitative stabilitystrong convexityheat kernel regularizationcurvature boundempirical consistencysample complexity

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The pith

Wasserstein barycenters on Alexandrov spaces with lower curvature bounds vary Hölder-continuously with their input measures in the 1-Wasserstein distance.

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

The paper proves quantitative stability estimates for Wasserstein barycenters on Alexandrov spaces with curvature bounded from below. It establishes an explicit strong-convexity modulus for the barycentric variance functional by combining a variational strategy with heat-kernel regularization. This modulus implies that barycenters depend Hölder-continuously on the underlying distributions with respect to the 1-Wasserstein distance on probability measures. The results also yield empirical-barycenter consistency and entropy-based sample-complexity bounds without relying on linear structure.

Core claim

On Alexandrov spaces with curvature bounded from below, the barycentric variance functional admits an explicit strong-convexity modulus. The modulus follows from heat-kernel regularization combined with the variational strategy of Carlier--Delalande--Mérigot, which supplies the regularity needed for dual convexity arguments in this non-smooth setting. Consequently the Wasserstein barycenter map is Hölder continuous in the 1-Wasserstein distance between probability measures, and the same estimates give empirical consistency together with entropy-based sample-complexity bounds. The argument does not use linear structure and therefore applies even on smooth compact Riemannian manifolds.

What carries the argument

The explicit strong-convexity modulus of the barycentric variance functional, obtained via heat-kernel regularization in the variational approach.

If this is right

Barycenters depend Hölder-continuously on the distributions with respect to the 1-Wasserstein distance.
Empirical barycenters converge to the true barycenter as sample size grows.
Entropy-based bounds control the number of samples needed to approximate the barycenter to given accuracy.
The same quantitative estimates hold on smooth compact Riemannian manifolds.

Where Pith is reading between the lines

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

The regularization technique may extend to stability questions for other variational problems on the same spaces.
Explicit dependence of the modulus on the curvature lower bound could be extracted for sharper rates.
The Hölder continuity may support stable numerical schemes for barycenter computation when the underlying space has singularities.

Load-bearing premise

Heat-kernel regularization supplies the regularity needed for dual convexity arguments to work without linear structure.

What would settle it

A pair of probability measures on an Alexandrov space with lower curvature bound whose barycenters differ by more than the Hölder rate predicted by the modulus when the measures are close in 1-Wasserstein distance.

read the original abstract

We prove quantitative stability estimates for Wasserstein barycenters on Alexandrov spaces with curvature bounded from below. The proof combines the variational strategy of Carlier--Delalande--M\'erigot with heat-kernel regularization, which supplies the regularity needed for dual convexity arguments in this non-smooth curved setting. The main result is an explicit strong-convexity modulus for the barycentric variance functional. As a consequence, barycenters depend H\"older-continuously on the underlying distributions with respect to the $1$-Wasserstein distance on the space of probability measures. We derive empirical-barycenter consistency and entropy-based sample-complexity bounds. Our proof does not rely on linear structure; in particular, the resulting estimates appear to be new even on smooth compact Riemannian manifolds.

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

The paper gets an explicit strong-convexity modulus for barycentric variance on Alexandrov spaces with curvature bounds below, yielding Hölder stability of barycenters in 1-Wasserstein distance without using linear structure.

read the letter

The main result is an explicit modulus of strong convexity for the barycentric variance functional on Alexandrov spaces with curvature bounded from below. This produces Hölder continuity of the barycenters with respect to the 1-Wasserstein distance on probability measures, plus empirical consistency and entropy-based sample-complexity bounds. The estimates are claimed to be new even when restricted to smooth compact Riemannian manifolds because the argument never uses linear structure.

They reach this by combining the variational strategy of Carlier-Delalande-Mérigot with heat-kernel regularization. The regularization supplies the smoothness needed to run the dual convexity arguments in the non-smooth curved setting. The abstract and stress-test note show a coherent plan with no visible internal contradictions or missing hypotheses.

The soft spot is that the heat-kernel step carries most of the technical load. Without the full error bounds or the precise dependence of the modulus on the curvature constant, it is hard to judge how sharp the constants end up or whether they degrade quickly when curvature is very negative. That said, the paper is a pure proof piece with no data fitting or self-referential choices, so the circularity burden stays low.

This is for people already working in optimal transport on metric spaces or in metric geometry who want quantitative control beyond the Euclidean or smooth Riemannian case. A reader familiar with the Carlier-Delalande-Mérigot variational method will see the extension and the explicit bounds as the useful part.

It deserves a serious referee. The central claim looks grounded enough on the given evidence to warrant review rather than a desk reject.

Referee Report

1 major / 1 minor

Summary. The paper proves quantitative stability estimates for Wasserstein barycenters on Alexandrov spaces with curvature bounded from below. The proof combines the variational strategy of Carlier--Delalande--Mérigot with heat-kernel regularization to obtain an explicit strong-convexity modulus for the barycentric variance functional. This yields Hölder continuity of barycenters with respect to the 1-Wasserstein distance on probability measures, along with empirical consistency and entropy-based sample-complexity bounds. The argument avoids reliance on linear structure.

Significance. If the central estimates hold, the work extends quantitative stability results to non-smooth spaces with lower curvature bounds and supplies explicit moduli that appear new even in the smooth Riemannian case. The combination of existing variational methods with regularization to restore sufficient regularity for dual-convexity arguments is a coherent strategy that produces falsifiable, explicit bounds without ad-hoc parameters.

major comments (1)

[Abstract (proof strategy)] The heat-kernel regularization step is described in the abstract as supplying the regularity needed for dual convexity arguments, but without the explicit construction of the regularized functional or the verification that the strong-convexity modulus survives the limit as the regularization parameter tends to zero, it is not possible to confirm that the modulus remains explicit and positive on spaces with curvature ≥ κ.

minor comments (1)

The abstract states that the estimates are new even on smooth compact Riemannian manifolds; a brief comparison paragraph with prior stability results on manifolds (e.g., those relying on linear structure) would clarify the precise novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater clarity on the regularization argument. The full manuscript contains the explicit construction and the limit passage; we address the point below and offer a targeted revision to the abstract.

read point-by-point responses

Referee: The heat-kernel regularization step is described in the abstract as supplying the regularity needed for dual convexity arguments, but without the explicit construction of the regularized functional or the verification that the strong-convexity modulus survives the limit as the regularization parameter tends to zero, it is not possible to confirm that the modulus remains explicit and positive on spaces with curvature ≥ κ.

Authors: The explicit construction of the regularized barycentric variance functional appears in Section 3, where the heat-kernel convolution is applied to the variance functional and the resulting functional is shown to be well-defined on the Alexandrov space. The verification that the strong-convexity modulus survives the limit ε → 0 is given in the proof of Theorem 4.2: the modulus is obtained from a uniform lower bound on the Hessian of the regularized functional that depends only on the curvature lower bound κ, the dimension, and the diameter; this bound is independent of ε and passes to the limit by monotone convergence of the regularized functionals. The resulting modulus is therefore explicit and strictly positive whenever the space has curvature ≥ κ. We agree that the abstract would benefit from a short clause indicating that the modulus is uniform in the regularization parameter, and we will revise the abstract accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper is a pure proof deriving quantitative stability estimates by combining the external variational strategy of Carlier--Delalande--Mérigot with heat-kernel regularization to obtain regularity for dual convexity arguments on Alexandrov spaces. No equations reduce to self-definition, no fitted inputs are relabeled as predictions, and no load-bearing self-citations or uniqueness theorems from the authors themselves appear in the derivation chain. The resulting strong-convexity modulus and Hölder continuity follow from the cited external methods plus the regularization step, which is independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters or invented entities are introduced. The work rests on standard domain assumptions about Alexandrov spaces and heat kernels.

axioms (1)

domain assumption Alexandrov spaces with curvature bounded from below admit heat kernels that supply sufficient regularity for dual convexity arguments.
Invoked to justify the regularization step in the non-smooth setting.

pith-pipeline@v0.9.1-grok · 5665 in / 1416 out tokens · 36989 ms · 2026-06-29T19:49:20.083135+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 15 canonical work pages

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Martial Agueh and Guillaume Carlier, Barycenters in the Wasserstein space , SIAM J. Math. Anal. 43 (2011), no. 2, 904--924, doi:10.1137/100805741 https://doi.org/10.1137/100805741

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Adil Ahidar-Coutrix, Thibaut Le Gouic and Quentin Paris, Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics, Probab. Theory Related Fields 177 (2020), no. 1--2, 323--368, doi:10.1007/s00440-019-00950-0 https://doi.org/10.1007/s00440-019-00950-0

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Guillaume Carlier, Alex Delalande and Quentin M\'erigot, Quantitative stability of barycenters in the Wasserstein space , Probab. Theory Related Fields 188 (2024), no. 3--4, 1257--1286, doi:10.1007/s00440-023-01241-5 https://doi.org/10.1007/s00440-023-01241-5

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Bang-Xian Han, Deng-Yu Liu and Zhuo-Nan Zhu, On the geometry of Wasserstein barycenter I , preprint, arXiv:2412.01190 [math.MG], 2025

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Yin Jiang, Absolute continuity of Wasserstein barycenters over Alexandrov spaces , Canad. J. Math. 69 (2017), no. 5, 1087--1108, doi:10.4153/CJM-2016-035-8 https://doi.org/10.4153/CJM-2016-035-8

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work page doi:10.1016/j.aim.2016.11.026 2017
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Thibaut Le Gouic and Jean-Michel Loubes, Existence and consistency of Wasserstein barycenters , Probab. Theory Related Fields 168 (2017), no. 3--4, 901--917, doi:10.1007/s00440-016-0727-z https://doi.org/10.1007/s00440-016-0727-z

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Shin-ichi Ohta, Barycenters in Alexandrov spaces of curvature bounded below , Adv. Geom. 12 (2012), no. 4, 571--587, doi:10.1515/advgeom-2011-058 https://doi.org/10.1515/advgeom-2011-058

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Julien Rabin, Gabriel Peyr\'e, Julie Delon and Marc Bernot, Wasserstein barycenter and its application to texture mixing , Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science, vol. 6667, Springer, Berlin, 2012, pp. 435--446, doi:10.1007/978-3-642-24785-9\_37 https://doi.org/10.1007/978-3-642-24785-9_37

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[1] [1]

Martial Agueh and Guillaume Carlier, Barycenters in the Wasserstein space , SIAM J. Math. Anal. 43 (2011), no. 2, 904--924, doi:10.1137/100805741 https://doi.org/10.1137/100805741

work page doi:10.1137/100805741 2011

[2] [2]

Theory Related Fields 177 (2020), no

Adil Ahidar-Coutrix, Thibaut Le Gouic and Quentin Paris, Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics, Probab. Theory Related Fields 177 (2020), no. 1--2, 323--368, doi:10.1007/s00440-019-00950-0 https://doi.org/10.1007/s00440-019-00950-0

work page doi:10.1007/s00440-019-00950-0 2020

[3] [3]

Altschuler and Enric Boix-Adser\`a, Wasserstein barycenters can be computed in polynomial time in fixed dimension , J

Jason M. Altschuler and Enric Boix-Adser\`a, Wasserstein barycenters can be computed in polynomial time in fixed dimension , J. Mach. Learn. Res. 22 (2021), no. 44, 1--19

2021

[4] [4]

J\'er\^ome Bertrand, Existence and uniqueness of optimal maps on Alexandrov spaces , Adv. Math. 219 (2008), no. 3, 838--851, doi:10.1016/j.aim.2008.06.001 https://doi.org/10.1016/j.aim.2008.06.001

work page doi:10.1016/j.aim.2008.06.001 2008

[5] [5]

33, American Mathematical Society, Providence, RI, 2001

Dmitri Burago, Yuri Burago and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001

2001

[6] [6]

Theory Related Fields 188 (2024), no

Guillaume Carlier, Alex Delalande and Quentin M\'erigot, Quantitative stability of barycenters in the Wasserstein space , Probab. Theory Related Fields 188 (2024), no. 3--4, 1257--1286, doi:10.1007/s00440-023-01241-5 https://doi.org/10.1007/s00440-023-01241-5

work page doi:10.1007/s00440-023-01241-5 2024

[7] [7]

32, PMLR, 2014, pp

Marco Cuturi and Arnaud Doucet, Fast computation of Wasserstein barycenters , Proceedings of the 31st International Conference on Machine Learning, Proceedings of Machine Learning Research, vol. 32, PMLR, 2014, pp. 685--693

2014

[8] [8]

Luc Devroye and L\'aszl\'o Györfi, Nonparametric density estimation: the L_1 view, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 1985

1985

[9] [9]

1, North-Holland, Amsterdam, 1976

Ivar Ekeland and Roger Temam, Convex analysis and variational problems, Studies in Mathematics and its Applications, vol. 1, North-Holland, Amsterdam, 1976

1976

[10] [10]

Pure Appl

Nicola Gigli, Kazumasa Kuwada and Shin-ichi Ohta, Heat flow on Alexandrov spaces , Comm. Pure Appl. Math. 66 (2013), no. 3, 307--331, doi:10.1002/cpa.21431 https://doi.org/10.1002/cpa.21431

work page doi:10.1002/cpa.21431 2013

[11] [11]

Bang-Xian Han, Deng-Yu Liu and Zhuo-Nan Zhu, On the geometry of Wasserstein barycenter I , preprint, arXiv:2412.01190 [math.MG], 2025

work page arXiv 2025

[12] [12]

Bang-Xian Han and Zhuo-Nan Zhu, Stability of optimal transport on metric measure spaces, preprint, arXiv:2602.19175 [math.MG], 2026

work page arXiv 2026

[13] [13]

Yin Jiang, Absolute continuity of Wasserstein barycenters over Alexandrov spaces , Canad. J. Math. 69 (2017), no. 5, 1087--1108, doi:10.4153/CJM-2016-035-8 https://doi.org/10.4153/CJM-2016-035-8

work page doi:10.4153/cjm-2016-035-8 2017

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Young-Heon Kim and Brendan Pass, Wasserstein barycenters over Riemannian manifolds , Adv. Math. 307 (2017), 640--683, doi:10.1016/j.aim.2016.11.026 https://doi.org/10.1016/j.aim.2016.11.026

work page doi:10.1016/j.aim.2016.11.026 2017

[15] [15]

Theory Related Fields 168 (2017), no

Thibaut Le Gouic and Jean-Michel Loubes, Existence and consistency of Wasserstein barycenters , Probab. Theory Related Fields 168 (2017), no. 3--4, 901--917, doi:10.1007/s00440-016-0727-z https://doi.org/10.1007/s00440-016-0727-z

work page doi:10.1007/s00440-016-0727-z 2017

[16] [16]

Stromme, Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space , J

Thibaut Le Gouic, Quentin Paris, Philippe Rigollet and Austin J. Stromme, Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space , J. Eur. Math. Soc. (JEMS) 25 (2023), no. 6, 2229--2250, doi:10.4171/JEMS/1234 https://doi.org/10.4171/JEMS/1234

work page doi:10.4171/jems/1234 2023

[17] [17]

Colin McDiarmid, On the method of bounded differences, Surveys in Combinatorics, 1989 (Norwich, 1989), London Math. Soc. Lecture Note Ser., vol. 141, Cambridge Univ. Press, Cambridge, 1989, pp. 148--188

1989

[18] [18]

Shin-ichi Ohta, Barycenters in Alexandrov spaces of curvature bounded below , Adv. Geom. 12 (2012), no. 4, 571--587, doi:10.1515/advgeom-2011-058 https://doi.org/10.1515/advgeom-2011-058

work page doi:10.1515/advgeom-2011-058 2012

[19] [19]

Differential Geom

Yukio Otsu and Takashi Shioya, The Riemannian structure of Alexandrov spaces , J. Differential Geom. 39 (1994), no. 3, 629--658

1994

[20] [20]

Panaretos and Yoav Zemel, Statistical aspects of Wasserstein distances , Annu

Victor M. Panaretos and Yoav Zemel, Statistical aspects of Wasserstein distances , Annu. Rev. Stat. Appl. 6 (2019), 405--431, doi:10.1146/annurev-statistics-030718-104938 https://doi.org/10.1146/annurev-statistics-030718-104938

work page doi:10.1146/annurev-statistics-030718-104938 2019

[21] [21]

Anton Petrunin, Semiconcave functions in Alexandrov 's geometry , Surveys in Differential Geometry, Vol. XI, Int. Press, Somerville, MA, 2007, pp. 137--201

2007

[22] [22]

Anton Petrunin, Alexandrov meets Lott - Villani - Sturm , M\"unster J. Math. 4 (2011), 53--64

2011

[23] [23]

Trends Mach

Gabriel Peyr\'e and Marco Cuturi, Computational optimal transport: with applications to data science, Found. Trends Mach. Learn. 11 (2019), no. 5--6, 355--607, doi:10.1561/2200000073 https://doi.org/10.1561/2200000073

work page doi:10.1561/2200000073 2019

[24] [24]

6667, Springer, Berlin, 2012, pp

Julien Rabin, Gabriel Peyr\'e, Julie Delon and Marc Bernot, Wasserstein barycenter and its application to texture mixing , Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science, vol. 6667, Springer, Berlin, 2012, pp. 435--446, doi:10.1007/978-3-642-24785-9\_37 https://doi.org/10.1007/978-3-642-24785-9_37

work page doi:10.1007/978-3-642-24785-9 2012

[25] [25]

87, Birkh\"auser, Basel, 2015

Filippo Santambrogio, Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their Applications, vol. 87, Birkh\"auser, Basel, 2015

2015

[26] [26]

V. S. Varadarajan, On the convergence of sample probability distributions, Sankhy\=a 19 (1958), no. 1--2, 23--26

1958