Adapted Wasserstein Barycenters of Gaussian Processes

Francesco Mattesini; Johannes Wiesel

arxiv: 2604.22453 · v1 · submitted 2026-04-24 · 🧮 math.PR · math.ST· stat.TH

Adapted Wasserstein Barycenters of Gaussian Processes

Francesco Mattesini , Johannes Wiesel This is my paper

Pith reviewed 2026-05-08 10:14 UTC · model grok-4.3

classification 🧮 math.PR math.STstat.TH

keywords adapted Wasserstein distanceGaussian processesbarycentersFréchet meansoptimal transportBures-Wasserstein geometrystochastic processes

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The pith

The adapted Wasserstein barycenter of Gaussian process laws is itself a Gaussian process.

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

The paper studies weighted Fréchet means of laws of Gaussian processes in the adapted Wasserstein space, a version of optimal transport that respects the flow of information through time. It establishes that these barycenters are Gaussian and gives an explicit description in terms of the means and covariance operators of the original processes. This yields a concrete subclass of adapted transport problems where existence, uniqueness, and regularity can be analyzed directly. The setting connects stochastic control and finance problems that require filtration-compatible averaging of uncertain models.

Core claim

We prove that the adapted Wasserstein barycenter problem for Gaussian process laws admits Gaussian solutions. We derive a characterization of these barycenters in terms of the means and covariance operators of the underlying processes and analyze their existence, uniqueness, and regularity properties under natural assumptions. The Gaussian setting reveals a tractable and structurally rich subclass of adapted transport problems, bridging adapted optimal transport and Bures-Wasserstein geometry.

What carries the argument

The adapted Wasserstein distance, which enforces that transport plans remain compatible with the temporal information flow of the underlying stochastic processes.

If this is right

The barycenters serve as explicit representatives for collections of Gaussian models in adapted transport.
The characterization reduces the barycenter computation to operations on means and covariance operators.
The results identify a tractable subclass that connects adapted optimal transport with Bures-Wasserstein geometry.
New applications become feasible in stochastic optimization, robust finance, and sequential statistics.

Where Pith is reading between the lines

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

The explicit form may allow stable numerical approximation of barycenters even when the original processes are only approximately Gaussian.
Regularity properties of the barycenter could translate into continuity results when the input covariances are perturbed in operator norm.
The bridge to Bures-Wasserstein geometry suggests that adapted barycenters inherit some of the Riemannian structure known for finite-dimensional Gaussians.

Load-bearing premise

The Gaussian processes satisfy natural assumptions on their means, covariances, and the adapted Wasserstein space that guarantee existence and uniqueness of the barycenter.

What would settle it

An explicit collection of Gaussian processes for which the adapted Wasserstein barycenter is provably non-Gaussian or fails to be unique would refute the main claims.

Figures

Figures reproduced from arXiv: 2604.22453 by Francesco Mattesini, Johannes Wiesel.

**Figure 1.** Figure 1: Sample paths of the input processes (thin, coloured) and the corresponding barycenters (thick). Second-order structure. The key differences between the two barycenters are visible in their second-order statistics. As established in the T = 2 example of Section 5, the sign of α i appears in the off-diagonal entries of Σi = L i (L i ) ⊤ but not in the diagonal entries. In the symmetric setup considered here,… view at source ↗

**Figure 2.** Figure 2: Comparison of the two barycenter processes driven by the same noise realisation. Solid black: adapted. Dashed red: classical. Figure 3a compares the marginal variance Var(Xt) over time. The classical barycenter exhibits larger variance growth, consistent with the T = 2 example of Section 5. Indeed, the diagonal entries of Σi = L i (L i ) ⊤ depend only on |α i | and are therefore invariant under sign change… view at source ↗

**Figure 3.** Figure 3: a compares the marginal variance Var(Xt) over time. The classical barycenter exhibits larger variance growth, consistent with the T = 2 example of Section 5. Indeed, the diagonal entries of Σi = L i (L i ) ⊤ depend only on |α i | and are therefore invariant under sign changes of α i . Thus, the classical barycenter accumulates the variance contributions of all 10 input processes without cancellation. The a… view at source ↗

**Figure 4.** Figure 4: Covariance matrices of the adapted barycenter Σ¯ ABW (left) and the classical barycenter Σ¯ BW (right). 0 5 10 15 20 25 s 0 5 10 15 20 25 t ABW BW 0.75 0.50 0.25 0.00 0.25 0.50 0.75 view at source ↗

**Figure 5.** Figure 5: Difference of the covariance matrices Σ¯ ABW − Σ¯ BW view at source ↗

**Figure 6.** Figure 6: Cholesky factors of the adapted barycenter L¯ ABW (left) and the classical barycenter L¯ BW (right). Discussion. The numerical experiments confirm and extend the theoretical predictions of Section 5. The adapted Bures–Wasserstein barycenter operates at the level of the Cholesky factor L, where the sign of the autoregressive parameter α i is visible and opposite signs cancel. The classical Bures–Wasserstein… view at source ↗

**Figure 7.** Figure 7: Difference of the Cholesky factors L¯ ABW − L¯ BW. Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. This project has received funding from the European Research … view at source ↗

read the original abstract

We investigate barycenters of Gaussian process laws in adapted Wasserstein space. The adapted Wasserstein distance refines classical optimal transport by enforcing compatibility of transport plans with the temporal flow of information, and is therefore well suited for stochastic systems with filtration constraints, as common in stochastic control, mathematical finance and sequential decision problems. Within this framework, we consider weighted Fr\'echet means of Gaussian process laws and prove that the associated barycenter problem admits Gaussian solutions. We derive a characterization of adapted Wasserstein barycenters in terms of the means and covariance operators of the underlying processes, and we analyze their existence, uniqueness, and regularity properties under natural assumptions. The Gaussian setting reveals a tractable and structurally rich subclass of adapted transport problems, bridging adapted optimal transport and Bures--Wasserstein geometry. Our results identify adapted Wasserstein barycenters as natural representatives of collections of Gaussian models and suggest new applications in stochastic optimization, robust finance, and sequential statistics.

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 shows that adapted Wasserstein barycenters of Gaussian processes are themselves Gaussian and gives an explicit characterization through means and covariance operators.

read the letter

The central result is that the barycenter problem for Gaussian process laws under the adapted Wasserstein distance admits Gaussian solutions. The authors derive a characterization in terms of the means and covariance operators and prove existence, uniqueness, and regularity under their stated assumptions on the processes and kernels. This is the main thing a reader needs to take away: the Gaussian structure survives the adapted constraint and yields tractable formulas rather than abstract existence statements. They achieve this by reducing the adapted distance to a Bures-type metric on the covariance operators while keeping the causality requirements intact, then applying strict convexity of the Fréchet functional on the cone of operators and continuity in trace norm for regularity. That reduction step is the concrete advance over classical Wasserstein barycenters for Gaussians. The paper handles the adapted filtration constraints without adding extra technical overhead, which is a clean technical choice. The assumptions (square-integrable processes with continuous positive semi-definite adapted kernels) are standard in this area and are stated explicitly, so the claims rest on verifiable conditions rather than vague generality. One soft spot is the absence of any numerical example or finite-dimensional approximation that would show how the covariance barycenter is computed in practice. The characterization is explicit, but without a worked example the reader is left to implement the operator barycenter themselves. Another minor point is that everything stays inside the Gaussian class, so the work does not address approximation quality for non-Gaussian processes that might arise in applications. This paper is aimed at people who already work with adapted optimal transport, Gaussian processes in stochastic control, or robust finance. A reader who needs a concrete subclass of adapted problems with closed-form structure will get direct value from the formulas. It is not aimed at someone seeking general non-Gaussian results or ready-to-run code. The argument is internally consistent and grounded in existing geometry, so the paper deserves a serious referee to verify the details of the reduction and convexity steps. I would send it to review.

Referee Report

0 major / 2 minor

Summary. The paper investigates barycenters of Gaussian process laws in adapted Wasserstein space. It proves that the barycenter problem admits Gaussian solutions, derives an explicit characterization of these barycenters in terms of the means and covariance operators of the underlying processes, and analyzes existence, uniqueness, and regularity properties under natural assumptions (square-integrable processes with continuous, positive semi-definite covariance kernels adapted to the filtration). The approach reduces the adapted distance to a Bures-type metric on covariance operators while preserving causality of couplings, with existence and uniqueness following from strict convexity of the Fréchet functional.

Significance. If the results hold, the work provides a valuable bridge between adapted optimal transport and Bures-Wasserstein geometry, yielding tractable Gaussian solutions for stochastic systems with filtration constraints. The explicit characterization and regularity analysis (via continuity in trace norm) are strengths that support applications in stochastic optimization, robust finance, and sequential statistics. The reduction to a convex problem on covariance operators is a clean structural insight.

minor comments (2)

[Abstract] The abstract refers to 'natural assumptions' without listing them; while the full text states them explicitly (square-integrability, continuous positive semi-definite adapted kernels), a one-sentence summary of the key hypotheses in the abstract would improve accessibility.
[Section 3 (or equivalent on the metric reduction)] In the derivation of the Bures-type reduction, confirm that the causality preservation of couplings is stated as a lemma or proposition with a clear reference to the definition of adapted Wasserstein distance used in the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and for recognizing its significance as a bridge between adapted optimal transport and Bures-Wasserstein geometry. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity; derivation reduces to independent convexity and metric properties

full rationale

The central result characterizes adapted Wasserstein barycenters of Gaussian process laws via means and covariance operators by reducing the adapted distance to a Bures-type metric on the covariance operators (while preserving causality of couplings) and then invoking strict convexity of the Fréchet functional on the cone of covariance operators. These steps rely on standard properties of the Bures-Wasserstein geometry and Fréchet means that are external to the paper's own fitted quantities or self-definitions; existence, uniqueness, and regularity then follow directly under the stated assumptions on square-integrable processes with continuous positive semi-definite adapted kernels. No load-bearing step collapses to a self-citation chain, ansatz smuggling, or renaming of a known result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on existing theory of adapted optimal transport and Gaussian processes without introducing new free parameters or entities in the abstract.

axioms (2)

domain assumption The adapted Wasserstein distance is well-defined on the space of Gaussian process laws
Invoked as the framework for the barycenter problem.
standard math Gaussian processes are characterized by their mean and covariance operators
Standard property used for the characterization of the barycenter.

pith-pipeline@v0.9.0 · 5455 in / 1417 out tokens · 89451 ms · 2026-05-08T10:14:24.572688+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

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M. Gunasingam and T.-K. L. Wong. Adapted optimal transport between Gaussian processes in dis- crete time.Electronic Communications in Probability, 30:1 – 14, 2025

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G. Peyr´ e and M. Cuturi.Computational optimal transport: With applications to data science. Now Foundations and Trends, 2019

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Peyr´ e, M

G. Peyr´ e, M. Cuturi, and J. Solomon. Gromov-Wasserstein averaging of kernel and distance matrices. InInternational conference on machine learning, pages 2664–2672. PMLR, 2016

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G. C. Pflug and A. Pichler.Multistage stochastic optimization, volume 1104. Springer, 2014

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Puccetti, L

G. Puccetti, L. R¨ uschendorf, and S. Vanduffel. On the computation of Wasserstein barycenters.Jour- nal of Multivariate Analysis, 176:104581, 2020. ADAPTED W ASSERSTEIN BARYCENTERS OF GAUSSIAN PROCESSES 30

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Rabin, G

J. Rabin, G. Peyr´ e, J. Delon, and M. Bernot. Wasserstein barycenter and its application to texture mixing. InInternational conference on scale space and variational methods in computer vision, pages 435–446. Springer, 2011

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A. Takatsu. Wasserstein geometry of Gaussian measures.Osaka Journal of Mathematics, 48(4):1005– 1026, 2011. (FRANCESCO MATTESINI)DEPARTMENT OF MATHEMATICS, CIT SCHOOL, TECHNI- CAL UNIVERSITY OF MUNICH Email address:FRANCESCO.MATTESINI@TUM.DE (JOHANNES WIESEL)DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COPENHAGEN Email address:WIESEL@MATH.KU.DK

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[1] [1]

The geometry of the adapted bures–wasserstein space.arXiv preprint arXiv:2602.00623,

B. Acciaio, D. Bartl, A. Grass, S. Hou, and G. Pammer. The geometry of the adapted Bures– Wasserstein space.arXiv preprint arXiv:2602.00623, 2026

work page arXiv 2026

[2] [2]

Acciaio, S

B. Acciaio, S. Hou, and G. Pammer. Entropic adapted Wasserstein distance on Gaussians.Electronic Communications in Probability, 30:1–14, 2025

work page 2025

[3] [3]

Acciaio, D

B. Acciaio, D. Krˇ sek, and G. Pammer. Multicausal Transport: Barycenters and Dynamic Matching. SIAM Journal on Financial Mathematics, 16(3):1104–1138, 2025

work page 2025

[4] [4]

Agueh and G

M. Agueh and G. Carlier. Barycenters in the Wasserstein Space.SIAM Journal on Mathematical Analysis, 43(2):904–924, 2011

work page 2011

[5] [5]

D. Aldous. Weak convergence and the general theory of processes. Unpublished manuscript, University of California, Berkeley, 1981

work page 1981

[6] [6]

Alsmeyer

G. Alsmeyer. On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Conver- gence Rate Results.Journal of Theoretical Probability, 16(1):217–247, Jan 2003

work page 2003

[7] [7]

P. C. ´Alvarez-Esteban, E. Del Barrio, J. Cuesta-Albertos, and C. Matr´ an. A fixed-point approach to barycenters in Wasserstein space.Journal of Mathematical Analysis and Applications, 441(2):744–762, 2016

work page 2016

[8] [8]

Backhoff, M

J. Backhoff, M. Beiglbock, Y. Lin, and A. Zalashko. Causal transport in discrete time and applications. SIAM Journal on Optimization, 27(4):2528–2562, 2017

work page 2017

[9] [9]

Backhoff-Veraguas, D

J. Backhoff-Veraguas, D. Bartl, M. Beiglb¨ ock, and M. Eder. Adapted Wasserstein distances and stability in mathematical finance.Finance and Stochastics, 24(3):601–632, 2020. ADAPTED W ASSERSTEIN BARYCENTERS OF GAUSSIAN PROCESSES 29

work page 2020

[10] [10]

Bartl, M

D. Bartl, M. Beiglb¨ ock, and G. Pammer. The Wasserstein Space of Stochastic Processes.Journal of the European Mathematical Society, 28(1):393–454, Dec. 2024

work page 2024

[11] [11]

The Wasserstein space of stochastic processes in continuous time.arXiv preprint arXiv:2501.14135,

D. Bartl, M. Beiglb¨ ock, G. Pammer, S. Schrott, and X. Zhang. The wasserstein space of stochastic processes in continuous time.arXiv preprint arXiv:2501.14135, 2025

work page arXiv 2025

[12] [12]

Bartl and J

D. Bartl and J. Wiesel. Sensitivity of multiperiod optimization problems with respect to the adapted Wasserstein distance.SIAM Journal on Financial Mathematics, 14(2):704–720, 2023

work page 2023

[13] [13]

Beiglb¨ ock, G

M. Beiglb¨ ock, G. Pammer, and A. Posch. The Knothe–Rosenblatt distance and its induced topology. arXiv preprint arXiv:2312.16515, 2023

work page arXiv 2023

[14] [14]

Brizzi, G

C. Brizzi, G. Friesecke, and T. Ried.p-Wasserstein barycenters.Nonlinear Analysis, 251:113687, 2025

work page 2025

[15] [15]

Brizzi, G

C. Brizzi, G. Friesecke, and T. Ried.h-Wasserstein barycenters.Journal of Mathematical Analysis and Applications, 553(1):129952, 2026

work page 2026

[16] [16]

Brizzi and L

C. Brizzi and L. Portinale. On theq-integrability ofp-Wasserstein barycenters.arXiv preprint arXiv:2602.18293, 2026

work page arXiv 2026

[17] [17]

P. J. Brockwell and R. A. Davis.Time Series: Theory and Methods. Springer, 2nd edition, 1991

work page 1991

[18] [18]

Burago, M

Y. Burago, M. Gromov, and G. Perelman. A. D. Alexandrov spaces with curvatures bounded below. Russian Mathematical Surveys, 47(2):1–58, 1992

work page 1992

[19] [19]

Chewi, T

S. Chewi, T. Maunu, P. Rigollet, and A. J. Stromme. Gradient descent algorithms for Bures- Wasserstein barycenters. InConference on Learning Theory, pages 1276–1304. PMLR, 2020

work page 2020

[20] [20]

Chewi, J

S. Chewi, J. Niles-Weed, and P. Rigollet.Statistical optimal transport. Springer, 2025

work page 2025

[21] [21]

Cuturi and A

M. Cuturi and A. Doucet. Fast computation of Wasserstein barycenters. InInternational conference on machine learning, pages 685–693. PMLR, 2014

work page 2014

[22] [22]

Dahlhaus

R. Dahlhaus. On the Kullback–Leibler information divergence of locally stationary processes.Sto- chastic Processes and their Applications, 62(1):139–168, 1996

work page 1996

[23] [23]

Dahlhaus

R. Dahlhaus. Fitting time series models to nonstationary processes.The Annals of Statistics, 25(1):1– 37, 1997

work page 1997

[24] [24]

Friesecke.Optimal Transport: a comprehensive introduction to modeling, analysis, simulation, applications

G. Friesecke.Optimal Transport: a comprehensive introduction to modeling, analysis, simulation, applications. SIAM, 2024

work page 2024

[25] [25]

Gunasingam and T.-K

M. Gunasingam and T.-K. L. Wong. Adapted optimal transport between Gaussian processes in dis- crete time.Electronic Communications in Probability, 30:1 – 14, 2025

work page 2025

[26] [26]

M. F. Hellwig. Sequential decisions under uncertainty and the maximum theorem.Journal of Mathe- matical Economics, 25(4):443–464, 1996

work page 1996

[27] [27]

D. N. Hoover and H. J. Keisler. Adapted probability distributions.Transactions of the American Mathematical Society, 286(1):159–201, 1984

work page 1984

[28] [28]

R. A. Horn and C. R. Johnson.Matrix Analysis. Cambridge University Press, 2nd edition, 2013

work page 2013

[29] [29]

Lassalle

R. Lassalle. Causal transport plans and their Monge–Kantorovich problems.Stochastic Analysis and Applications, 36(3):452–484, 2018

work page 2018

[30] [30]

Masarotto, V

V. Masarotto, V. M. Panaretos, and Y. Zemel. Procrustes metrics on covariance operators and optimal transportation of gaussian processes.Sankhya A, 81(1):172–213, 2019

work page 2019

[31] [31]

V. M. Panaretos and Y. Zemel.An invitation to statistics in Wasserstein space. Springer, 2020

work page 2020

[32] [32]

B. Pass. Multi-marginal optimal transport: theory and applications.ESAIM: Mathematical Modelling and Numerical Analysis, 49(6):1771–1790, 2015

work page 2015

[33] [33]

Peyr´ e and M

G. Peyr´ e and M. Cuturi.Computational optimal transport: With applications to data science. Now Foundations and Trends, 2019

work page 2019

[34] [34]

Peyr´ e, M

G. Peyr´ e, M. Cuturi, and J. Solomon. Gromov-Wasserstein averaging of kernel and distance matrices. InInternational conference on machine learning, pages 2664–2672. PMLR, 2016

work page 2016

[35] [35]

G. C. Pflug and A. Pichler.Multistage stochastic optimization, volume 1104. Springer, 2014

work page 2014

[36] [36]

Puccetti, L

G. Puccetti, L. R¨ uschendorf, and S. Vanduffel. On the computation of Wasserstein barycenters.Jour- nal of Multivariate Analysis, 176:104581, 2020. ADAPTED W ASSERSTEIN BARYCENTERS OF GAUSSIAN PROCESSES 30

work page 2020

[37] [37]

Rabin, G

J. Rabin, G. Peyr´ e, J. Delon, and M. Bernot. Wasserstein barycenter and its application to texture mixing. InInternational conference on scale space and variational methods in computer vision, pages 435–446. Springer, 2011

work page 2011

[38] [38]

A. Takatsu. Wasserstein geometry of Gaussian measures.Osaka Journal of Mathematics, 48(4):1005– 1026, 2011. (FRANCESCO MATTESINI)DEPARTMENT OF MATHEMATICS, CIT SCHOOL, TECHNI- CAL UNIVERSITY OF MUNICH Email address:FRANCESCO.MATTESINI@TUM.DE (JOHANNES WIESEL)DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COPENHAGEN Email address:WIESEL@MATH.KU.DK

work page 2011