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Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimization

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Pith's one-line read WDHA computes the unregularized Wasserstein barycenter in O(m log m) per-potential-update time and O(m) memory, with O(1/T) convergence to a stationary point under a strong-convexity-smoothness projection.

desk verdict Promising new descent-ascent recipe for unregularized barycenters, with a theory-practice gap the paper itself admits. read the letter →

arxiv 2501.14635 v2 pith:M7MFXDPD submitted 2025-01-24 stat.ML cs.LG

Kaheon Kim , Rentian Yao , Changbo Zhu , Xiaohui Chen This is my paper · ORCID
classification stat.MLcs.LG MSC 49Q2265K1090C26
keywords Wassersteinbarycenteroptimaltransportnonconvex-concaveminimaxprimal-dualalgorithmhomogeneousSobolevgradientKantorovichpotentialentropicregularizationimagedensityaveraging
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the unregularized Wasserstein barycenter—the natural average of probability densities in optimal-transport geometry—can be computed by a primal-dual coordinate method that avoids both the O($m^{3}$) cost of solving transport maps and the blur introduced by entropic regularization. The proposed WDHA algorithm alternates a Wasserstein-gradient descent on the barycenter density with a homogeneous Sobolev ($\dot H^1$) gradient ascent on the dual potential functions, reaching O(m log m) time and O(m) memory per potential update on an m-point grid. Under the assumption that the dual potentials are restricted to the class of α-strongly convex, β-smooth functions, the paper proves that the average squared Wasserstein gradient decays to zero at rate O(1/T). If the claim holds, averaging high-resolution image-type densities becomes practical without the blurring trade-off of entropically regularized barycenter methods.

What carries the argument

The machinery is the pairing of two optimization geometries for the two blocks of the minimax objective. For the dual block, the Kantorovich functional $I^\mu_\nu$ is maximized over potentials in the class $F_{\alpha,\beta}$ of $\alpha$-strongly convex, $\beta$-smooth functions using the $\dot H^1$ gradient, whose defining property is the isometric embedding $\|\nabla \phi\|_{L^2}^2 = \|\nu - \mu\|_{\dot H^{-1}}^2$. For the primal block, the barycenter is updated by the Wasserstein gradient $\nabla_W J(\nu,\phi) = \mathrm{id} - \frac{1}{n}\sum_i \nabla\phi_i$, which is the identity map minus the average gradient of the potentials. The alternation works because the $\dot H^1$ ascent can be computed in $O(m\log m)$ time and $O(m)$ space via the fast Legendre transform and the inverse Laplacian, and because the Wasserstein descent interprets the average potential gradient as a transport map. Projection onto $F_{\alpha,\beta}$ is what makes the functional strongly concave in $\phi$ (Lemma 3.1), giving the contraction $\|\phi_{t+1} - \tilde\phi_\nu\|_{\dot H^1}^2 \le (1 - A\eta)\|\phi_t - \tilde\phi_\nu\|_{\dot H^1}^2$ that drives the proof of Theorem 3.6.

What would settle it

Run the implemented Algorithm 4 on a smooth 2D example with a known barycenter and monitor whether the average squared Wasserstein gradient norm decays as O(1/T) under the step-size window prescribed in Theorem 3.6, and whether the iterate densities remain uniformly bounded. A violation of either condition would demonstrate that the theorem does not cover the practical algorithm; equivalently, exhibit two distributions for which the strong-concavity inequality of Lemma 3.1 fails for potentials produced by the second-convex-conjugate update.

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Extended reading notes

Core claim

The central claim is that the Wasserstein barycenter problem, written as $\min_\nu \max_{\phi_i \in F_{\alpha,\beta}} \frac{1}{n} \sum_i I^{\mu_i}_\nu(\phi_i)$, is a nonconvex-concave minimax problem whose stationary points can be reached by alternating two geometrically matched gradient steps: a Wasserstein descent $\nu_{t+1} = (\mathrm{id} - \tau_t \nabla_W J(\nu_t, \phi_t))_\# \nu_t$ on the barycenter, and a $\dot H^1$ ascent on each Kantorovich potential using $\nabla_{\phi_i} I^{\mu_i}_\nu = (-\Delta)^{-1}(-\nu + (\nabla \phi_i^*)_\# \mu_i)$, followed by projection onto $F_{\alpha,\beta}$. The paper proves in Theorem 3.6 that with step sizes $\eta < 1/B$ and $\tau$ suitably small, the average squared $L^2$-norm of the Wasserstein gradient of the barycenter functional vanishes at rate $O(1/T)$, and that each inner update costs $O(m\log m)$ time and $O(m)$ space. Empirically, the paper shows that the implemented variant—where projection is replaced by the second convex conjugate $(\cdot)^{**}$—produces sharp, unblurred barycenters on $1024 \times 1024$ synthetic and handwritten-digit images, with lower objective values than entropy-regularized baselines. The paper is explicit that the convergence theorem applies to the projected variant; Section 3.5 states that for the implemented second-conjugate variant the strong-concavity lemma no longer holds and the inner functional is only guaranteed to be concave.

Load-bearing premise

The convergence proof holds only for the variant that projects dual potentials onto the class of strongly convex, smooth functions, while the algorithm actually run replaces that projection with a cheaper one that only enforces convexity; if that replacement breaks the strong-concavity lemma, the headline complexity result has no proven guarantee for the practical method.

Editorial extensions

If this is right

  • Each Kantorovich potential update costs O(m log m) time and O(m) memory, against O(m^3) time and O(m^2) memory for grid linear-programming transport solvers, so high-resolution 2D and 3D densities become computationally feasible.
  • With the step-size rules of Theorem 3.6, the algorithm reaches an ε-stationary point of the barycenter functional in O(1/ε^2) iterations, matching the rate of Euclidean nonconvex-concave gradient descent-ascent methods.
  • Because WDHA targets the unregularized barycenter, its output does not carry the entropy bias of regularized methods; on the paper's examples the barycentric edges remain sharp and the barycenter functional values are lower.
  • The algorithm removes the per-iteration need to solve n optimal transport problems exactly; the transport maps are encoded implicitly through the updated dual potentials.
  • The approach is presently limited to barycenters of 2D or 3D distributions supported on a compact domain.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The split between the theory (projected Algorithm 3) and the implementation (second-conjugate Algorithm 4) suggests a concrete route to a rigorous O(m log m) guarantee: either prove that stability and strong-concavity-type properties survive the second-convex-conjugate projection, or design an O(m log m) projection onto F_{α,β}. This is the paper's own open gap, stated in Section 3.5.
  • The same Wasserstein-descent / Hdot1-ascent alternation is not specific to barycenters; it should extend to any saddle-point problem whose inner maximization is a Kantorovich dual, such as multimarginal transport, unbalanced optimal transport, or weighted barycenters, provided the Hdot1 isometry holds for each dual block.
  • The sharp-edge advantage over entropy-regularized methods is likely most pronounced for piecewise-constant or low-regularity target densities; for smooth densities, regularized methods at matched runtime may achieve comparable functional values, so the practical edge is probably not universal.
  • The paper's bounded-density assumption (max_t ||ν_t||_∞ ≤ V) is flagged by the authors as unproven; a counterexample with iterates developing unbounded density spikes would invalidate Theorem 3.6 even for the projected Algorithm 3.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper proposes the Wasserstein-Descent Ḣ1-Ascent (WDHA) algorithm for computing unregularized Wasserstein barycenters of densities on a grid. The barycenter functional is recast as a nonconvex-concave minimax problem over ν and n Kantorovich potentials in the class F_{α,β} of α-strongly convex, β-smooth functions. The algorithm alternates a Wasserstein-gradient descent step on the barycenter ν with Ḣ1-gradient ascent steps on the potentials, followed by projection. The main theoretical result (Theorem 3.6) states that, under density bounds and a uniform boundedness assumption on the iterates, the averaged squared Wasserstein gradient of F_{α,β} converges at O(1/T). The authors claim O(m log m) per-iteration time and O(m) space for a practical variant, Algorithm 4, which replaces the F_{α,β} projection by the second convex conjugate. Numerical comparisons on 1024×1024 two-dimensional images show sharper and less blurred barycenters than entropic baselines.

Significance. Computing exact Wasserstein barycenters at near-linear cost is an important open problem, and a correct algorithm with O(m log m) per-iteration cost and O(1/T) stationarity would be a significant advance over entropic Sinkhorn-type methods. The paper's formulation is clean, and its reliance on the back-and-forth Ḣ1 machinery of Jacobs and Léger is appropriate and cited. The authors are also transparent about the gap between the analyzed projected algorithm and the implemented convex-conjugate version, and they include a 1D ground-truth comparison of the two variants. However, because the central complexity claim is tied to the variant whose convergence is not proven, the contribution as stated is not yet established.

major comments (3)
  1. [Section 3.5, Theorem 3.6, Lemma B.2] The convergence analysis (Theorem 3.6 and its supporting Lemma B.2) is proved for Algorithm 3, whose projection onto F_{α,β} costs O(m^2) (as the paper notes, citing Simonetto 2021). The implemented and tested method, Algorithm 4, replaces this projection with the second convex conjugate, which only enforces convexity. The paper itself states in Section 3.5 that 'Lemma 1 no longer holds, and I is only guaranteed to be concave' for Algorithm 4. Consequently, the constants A and B in Lemma 3.1 degenerate (A=0, B=∞), the contraction factor (1−Aη) in Lemma B.2 becomes vacuous, and the O(1/T) bound on the averaged squared gradient of F_{α,β} does not apply to Algorithm 4. Moreover, Algorithm 4's iterates need not remain in F_{α,β}, so the stationarity measure in Theorem 3.6 is mismatched. The headline claim of a nearly-linear-time algorithm with a convergence guarantee therefore refers to two different algorithms, and the gap is not resolved.
  2. [Section 3.4, Remark 3.7(iii), abstract] The theorem establishes convergence only to a stationary point of the restricted functional F_{α,β}, not of the original barycenter functional F. The equivalence to a true Wasserstein barycenter requires the additional, unverified condition that the Kantorovich potentials between the barycenter and each μ_i belong to F_{α,β} (Remark 3.7(iii)). For the discretized setting and for Algorithm 4, this condition is neither established nor tested. Thus the abstract's phrase 'computing the exact barycenter' is stronger than what Theorem 3.6 supports.
  3. [Section 3.5, Appendix D.2] The analysis is carried out in the infinite-dimensional continuous setting, while the experiments and the complexity claim concern a fixed m-point grid with finite-difference gradients, a discrete convex conjugate routine, and a heuristic scattered-data pushforward. No discretization error bounds are provided, and the assumptions of Theorem 3.6 (density bounds 0<a≤μ_i≤b, uniform bound sup_t ||ν_t||∞≤V, and projection onto F_{α,β}) are not verified for the discrete implementation. A practical reader therefore cannot conclude that the discrete algorithm satisfies the theorem's hypotheses.
minor comments (6)
  1. [Section 3.1] Typo: 'theoretically trackable' should be 'theoretically tractable'.
  2. [Section 3.3] Typo: 'We now dicuss' should be 'We now discuss'.
  3. [Section 4] Typo: 'Sythentic Uniform Distributions' should be 'Synthetic Uniform Distributions'.
  4. [Appendix C.2] Typos: 'barycernter' should be 'barycenter', and 'For each repetitiont' should be 'For each repetition t'.
  5. [Appendix A (Notations)] The entry 'F_{α,β} average of maximal functionals defined as F_{α,β} = 1/n Σ L^{μ_i}' uses the same symbol F_{α,β} for the function class and the averaged functional; for clarity, denote the averaged functional by F_{α,β}(ν) or another symbol.
  6. [Appendix B.5] In the proof of Lemma B.2, the notation I(φ_t) appears without superscripts; it should be I^{μ}_{ν}(φ_t) for consistency.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof chain is built on independent prior OT results and explicit assumptions, and the Algorithm 3/4 gap is an unproven-claim issue rather than a reduction of the conclusion to its inputs.

full rationale

The paper's convergence claim is a theorem about Algorithm 3, proved from stated assumptions (bounded densities, uniformly bounded iterates, potentials in F_{α,β}, and step-size conditions). The proof uses Lemma 3.1 and Lemma B.2, which are derived within the paper from properties of convex conjugation and the independent H1-gradient formula of Jacobs and Leger, not from the theorem being proved. No parameter is fitted to data and then reported as a prediction: the step sizes in the experiments are heuristic optimization hyperparameters, and the functional values are evaluated by the back-and-forth estimator of Jacobs and Leger, an external method. The self-citations (Zhu and Muller; Jiang, Zhu and Shao) appear only as background applications and are not load-bearing for the algorithm or the rates. The paper explicitly acknowledges the main weakness in Section 3.5: replacing the projection P_{F_{α,β}} with double convex conjugation means 'Lemma 1 no longer holds, and I is only guaranteed to be concave,' and the O(m log m) complexity is claimed for Algorithm 4 while the theorem covers Algorithm 3. This is a theory-practice gap and a correctness risk concerning the headline claim, but it is not a circular derivation: the statement 'Algorithm 4 has no proven convergence guarantee' is not equivalent to the algorithm's inputs or to a self-citation. Similarly, the restriction of potentials to F_{α,β} means the computed stationary point of F_{α,β} is an exact barycenter only under the additional assumption that the true Kantorovich potentials lie in that class, which is a stated assumption rather than a definitional identity. I therefore find no step in the derivation chain that reduces by construction to its own inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard optimal transport theory, density boundedness assumptions, and the technical boundedness of all iterates. The paper introduces no new physical entity, force, or conserved quantity. The main free parameters are the regularization class parameters and the heuristic step-size schedules; neither is fitted to the ground truth in a way that forces the empirical conclusions. The weakest structural assumption is the transferability of convergence from the projected Algorithm 3 to the implemented Algorithm 4, which is not proved.

free parameters (2)
  • alpha, beta (F_{alpha,beta} class parameters) = alpha=1e-3, beta=1e3 in 1D experiments; not used by Algorithm 4
    These define the strongly convex and smooth potential class in the theory. The convergence theorem applies only to this class, but the implemented Algorithm 4 does not enforce these bounds.
  • Step size schedules (eta_t, tau_t) = tau_t = exp(-t/T); eta starts at 0.05 or 0.5 and decays by 0.99 or 0.95 when the dual objective decreases
    The step sizes are chosen heuristically, not derived from the constants in Theorem 3.6. The theorem requires specific bounds on eta and tau that depend on unknown constants a, b, alpha, beta, and V.
assumptions (5)
  • domain assumption The input densities mu_i are absolutely continuous and satisfy 0 < a <= mu_i(x) <= b < infinity for all x in Omega.
    This is the standing assumption for the strong concavity and smoothness of the dual functional in Lemma 3.1, and it is stated before the main theorems.
  • domain assumption The iterates nu_t have uniformly bounded L-infinity norms, maxt ||nu_t||_inf <= V < infinity.
    Assumed for the convergence theorem in Theorem 3.6; the paper Remark 3.7(ii) notes this is an open question whether it follows from other assumptions.
  • standard math Brenier's theorem and the Kantorovich dual formulation hold for the regularized class F_{alpha,beta}.
    Used to justify that the dual problem evaluates W2^2 and that the Wasserstein gradient is id - grad(phi). Standard results in optimal transport (Santambrogio, 2015).
  • standard math Theorem 5.34 of Santambrogio (2015), relating the H^{-1} norm to the W2 distance with constants depending on L-infinity bounds.
    Used in Lemmas 3.4 and 3.5 and in the proof of Theorem 3.6 to bound changes in the optimal potentials in terms of W2.
  • standard math The projection onto F_{alpha,beta} is well-defined and unique, and the 1D Legendre-Fenchel transform and the pushforward computations of Jacobs and Leger run in O(m log m).
    Used in the computational complexity claims and in the proof of Lemma B.2. The O(m log m) bound is transferred from Jacobs and Leger (2020).

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Cite this review

Pith. "Pith review of Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimization." pith.science (2026). https://pith.science/paper/M7MFXDPD

@misc{pith2026250114635,
  author       = {Pith},
  title        = {Pith review of: Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7MFXDPD}},
  note         = {Machine review of arXiv:2501.14635}
}
abstract

The optimal transport barycenter (a.k.a. Wasserstein barycenter) is a fundamental notion of averaging that extends from the Euclidean space to the Wasserstein space of probability distributions. Computation of the unregularized barycenter for discretized probability distributions on point clouds is a challenging task when the domain dimension $d > 1$. Most practical algorithms for approximating the barycenter problem are based on entropic regularization. In this paper, we introduce a nearly linear time $O(m \log{m})$ and linear space complexity $O(m)$ primal-dual algorithm, the Wasserstein-Descent $\dot{\mathbb{H}}^1$-Ascent (WDHA) algorithm, for computing the exact barycenter when the input probability density functions are discretized on an $m$-point grid. The key success of the WDHA algorithm hinges on alternating between two different yet closely related Wasserstein and Sobolev optimization geometries for the primal barycenter and dual Kantorovich potential subproblems. Under reasonable assumptions, we establish the convergence rate and iteration complexity of WDHA to its stationary point when the step size is appropriately chosen. Superior computational efficacy, scalability, and accuracy over the existing Sinkhorn-type algorithms are demonstrated on high-resolution (e.g., $1024 \times 1024$ images) 2D synthetic and real data.

Figures

Figures reproduced from arXiv: 2501.14635 by the authors.

Figure 1
Figure 1. Illustration of Wasserstein barycenters computed by different methods. The goal is to compute the barycenter of four uniform densities supported on the square, circle, heart, and cross, respectively, as displayed in the top left image. The blended shape shown in the top middle image is the barycentric density computed using our method. Barycentric densities computed using CWB and DSB with reg = 0.005, and their thre… view at source ↗
Figure 2
Figure 2. Top row displays three exemplary digit 8 images. Bottom row displays barycenters computed by different methods using 300 iterations. smaller than the threshold such that the removed intensities amount to 10% of the total mass. The thresholded barycen￾ters are shown in the bottom left and right images in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Illustration of Wasserstein barycenters computed using WDHA, CWB and DSB. C.3. Experiments on 3D distributions We extended our WDHA implementation to handle 3D distributions and conducted an example in such setting. The goal is to compute the barycenter of three uniform densities supported on different shapes of ellipsoid contained in [0, 1]3 . Our simulation setup is: (x−0.5)2 a2 + (y−0.5)2 b 2 + (z−0.5)2 c 2 ≤ 0.8… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Illustration of Wasserstein barycenters computed using WDHA for 3D distributions. The results are smoothed using a Gaussian filter. Algorithm 5 L 2 -Descent H˙ 1 -Ascent Algorithm Initialize ν 1 , φ1 ; for t = 1, 2, · · · , T − 1 do for i = 1, 2, . . . , n do φb t+1 i …
Figure 5
Figure 5. Figure 5: Plot of output from Algorithm C.4 to uniform distributions on round disks. D. Additional Details D.1. Wasserstein gradient To compute ∇∇J (ν, φ), we apply the definition in subsection 2.3. Note that d dεJ (ν + εχ, ϕ) [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

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Works this paper leans on

47 extracted references · 39 canonical work pages

  1. [1]

    and Carlier, G

    Agueh, M. and Carlier, G. Barycenters in the W asserstein space. SIAM Journal on Mathematical Analysis, 43 0 (2): 0 904--924, 2011

    2011

  2. [2]

    C., Del Barrio, E., Cuesta-Albertos, J., and Matr \'a n, C

    \'A lvarez-Esteban, P. C., Del Barrio, E., Cuesta-Albertos, J., and Matr \'a n, C. A fixed-point approach to barycenters in wasserstein space. Journal of Mathematical Analysis and Applications, 441 0 (2): 0 744--762, 2016

    work page 2016

  3. [3]

    Gradient flows: in metric spaces and in the space of probability measures

    Ambrosio, L., Gigli, N., and Savar \'e , G. Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media, 2008

    work page 2008

  4. [4]

    and Rosenthal, J

    Beaulac, C. and Rosenthal, J. S. Introducing a new high-resolution handwritten digits data set with writer characteristics. SN Computer Science, 4 0 (1): 0 66, 2022

    work page 2022

  5. [5]

    Berman, R. J. Convergence rates for discretized M onge-- A mp \`e re equations and quantitative stability of optimal transport. Foundations of Computational Mathematics, 21 0 (4): 0 1099--1140, 2021

    work page 2021

  6. [6]

    and Rapaport, A

    Bernhard, P. and Rapaport, A. On a theorem of D anskin with an application to a theorem of V on N eumann- S ion. Nonlinear Analysis: Theory, Methods & Applications, 24 0 (8): 0 1163--1181, 1995

    work page 1995

  7. [7]

    and Castanon, D

    Bertsekas, D. and Castanon, D. The auction algorithm for the transportation problem. Annals of Operations Research, 20 0 (1): 0 67--96, 1989

    work page 1989

  8. [8]

    Bertsekas, D. P. Nonlinear programming. Journal of the Operational Research Society, 48 0 (3): 0 334--334, 1997

    work page 1997

Show all 47 references
  1. [9]

    Penalization of barycenters in the W asserstein space

    Bigot, J., Cazelles, E., and Papadakis, N. Penalization of barycenters in the W asserstein space. SIAM Journal on Mathematical Analysis, 51 0 (3): 0 2261--2285, 2019

    2019

  2. [10]

    On the linear convergence of the multimarginal S inkhorn algorithm

    Carlier, G. On the linear convergence of the multimarginal S inkhorn algorithm. SIAM Journal on Optimization, 32 0 (2): 0 786--794, 2022

    2022

  3. [11]

    Numerical methods for matching for teams and wasserstein barycenters

    Carlier, G., Oberman, A., and Oudet, E. Numerical methods for matching for teams and wasserstein barycenters. ESAIM: Mathematical Modelling and Numerical Analysis, 49 0 (6): 0 1621--1642, 2015

    2015

  4. [12]

    Entropic- W asserstein barycenters: PDE characterization, regularity, and CLT

    Carlier, G., Eichinger, K., and Kroshnin, A. Entropic- W asserstein barycenters: PDE characterization, regularity, and CLT . SIAM Journal on Mathematical Analysis, 53 0 (5): 0 5880--5914, 2021

    2021

  5. [13]

    Wasserstein regression

    Chen, Y., Lin, Z., and Müller, H.-G. Wasserstein regression. Journal of the American Statistical Association, 118 0 (542): 0 869--882, 2023

    2023

  6. [14]

    Chewi, S., Maunu, T., Rigollet, P., and Stromme, A. J. Gradient descent algorithms for B ures- W asserstein barycenters. In Conference on Learning Theory, pp.\ 1276--1304, 2020

    2020

  7. [15]

    Doubly regularized entropic W asserstein barycenters

    Chizat, L. Doubly regularized entropic W asserstein barycenters. arXiv preprint arXiv:2303.11844, 2023

    2023 arXiv

  8. [16]

    Fast L egendre-- F enchel transform and applications to H amilton-- J acobi equations and conservation laws

    Corrias, L. Fast L egendre-- F enchel transform and applications to H amilton-- J acobi equations and conservation laws. SIAM Journal on Numerical Analysis, 33 0 (4): 0 1534--1558, 1996

    1996

  9. [17]

    and Doucet, A

    Cuturi, M. and Doucet, A. Fast computation of W asserstein barycenters. In Proceedings of the 31st International Conference on Machine Learning, pp.\ 685--693, 2014

    2014

  10. [18]

    and M \"u ller, H.-G

    Dubey, P. and M \"u ller, H.-G. Fr \'e chet change-point detection. The Annals of Statistics, 48 0 (6): 0 3312--3335, 2020

    2020

  11. [19]

    Evans, L. C. Partial differential equations, volume 19. American Mathematical Society, 2022

    2022

  12. [20]

    Z., Boisbunon, A., Chambon, S., Chapel, L., Corenflos, A., Fatras, K., Fournier, N., Gautheron, L., Gayraud, N

    Flamary, R., Courty, N., Gramfort, A., Alaya, M. Z., Boisbunon, A., Chambon, S., Chapel, L., Corenflos, A., Fatras, K., Fournier, N., Gautheron, L., Gayraud, N. T., Janati, H., Rakotomamonjy, A., Redko, I., Rolet, A., Schutz, A., Seguy, V., Sutherland, D. J., Tavenard, R., Ton...

    2021

  13. [21]

    Fast optimal transport averaging of neuroimaging data

    Gramfort, A., Peyr \'e , G., and Cuturi, M. Fast optimal transport averaging of neuroimaging data. In Information Processing in Medical Imaging, pp.\ 261--272, 2015

    2015

  14. [22]

    and Rigollet, P

    H \"u tter, J.-C. and Rigollet, P. Minimax estimation of smooth optimal transport maps . The Annals of Statistics, 49 0 (2): 0 1166 -- 1194, 2021

    2021

  15. [23]

    and L \'e ger, F

    Jacobs, M. and L \'e ger, F. A fast approach to optimal transport: The back-and-forth method. Numerische Mathematik, 146 0 (3): 0 513--544, 2020

    2020

  16. [24]

    Debiased S inkhorn barycenters

    Janati, H., Cuturi, M., and Gramfort, A. Debiased S inkhorn barycenters. In Proceedings of the 37th International Conference on Machine Learning, pp.\ 4692--4701, 2020

    2020

  17. [25]

    Two-sample and change-point inference for non-euclidean valued time series

    Jiang, F., Zhu, C., and Shao, X. Two-sample and change-point inference for non-euclidean valued time series. Electronic Journal of Statistics, 18 0 (1): 0 848--894, 2024

    2024

  18. [26]

    Continuous wasserstein-2 barycenter estimation without minimax optimization

    Korotin, A., Li, L., Solomon, J., and Burnaev, E. Continuous wasserstein-2 barycenter estimation without minimax optimization. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=3tFAs5E-Pe

    2021

  19. [27]

    Wasserstein iterative networks for barycenter estimation

    Korotin, A., Egiazarian, V., Li, L., and Burnaev, E. Wasserstein iterative networks for barycenter estimation. In Advances in Neural Information Processing Systems, volume 35, pp.\ 15672--15686. Curran Associates, Inc., 2022

    2022

  20. [28]

    Kuhn, H. W. The H ungarian method for the assignment problem. Naval Research Logistics Quarterly, 2 0 (1-2): 0 83--97, 1955

    1955

  21. [29]

    Continuous regularized W asserstein barycenters

    Li, L., Genevay, A., Yurochkin, M., and Solomon, J. Continuous regularized W asserstein barycenters. In Proceedings of the 34th International Conference on Neural Information Processing Systems, pp.\ 17755--17765, 2020 a

    2020

  22. [30]

    Li, L., Genevay, A., Yurochkin, M., and Solomon, J. M. Continuous regularized wasserstein barycenters. Advances in Neural Information Processing Systems, 33: 0 17755--17765, 2020 b

    2020

  23. [31]

    On gradient descent ascent for nonconvex-concave minimax problems

    Lin, T., Jin, C., and Jordan, M. On gradient descent ascent for nonconvex-concave minimax problems. In International Conference on Machine Learning, pp.\ 6083--6093, 2020

    2020

  24. [32]

    Lin, T., Ho, N., Cuturi, M., and Jordan, M. I. On the complexity of approximating multimarginal optimal transport. The Journal of Machine Learning Research, 23 0 (1): 0 2835--2877, 2022

    2022

  25. [33]

    Luenberger, D. G. and Ye, Y. Linear and Nonlinear Programming. Springer New York, 2008

    2008

  26. [34]

    Plugin estimation of smooth optimal transport maps

    Manole, T., Balakrishnan, S., Niles-Weed, J., and Wasserman, L. Plugin estimation of smooth optimal transport maps. The Annals of Statistics, 52 0 (3): 0 966--998, 2024

    2024

  27. [35]

    and Pegon, P

    Nenna, L. and Pegon, P. Convergence rate of entropy-regularized multi-marginal optimal transport costs. Canadian Journal of Mathematics, pp.\ 1–21, 2024

    2024

  28. [36]

    Regularity as regularization: Smooth and strongly convex brenier potentials in optimal transport

    Paty, F.-P., d’Aspremont, A., and Cuturi, M. Regularity as regularization: Smooth and strongly convex brenier potentials in optimal transport. In International Conference on Artificial Intelligence and Statistics, pp.\ 1222--1232, 2020

    2020

  29. [37]

    and Cuturi, M

    Peyr \'e , G. and Cuturi, M. Computational optimal transport: With applications to data science. Foundations and Trends in Machine Learning , 11 0 (5-6): 0 355--607, 2019

    2019

  30. [38]

    Wasserstein barycenter and its application to texture mixing

    Rabin, J., Peyr \'e , G., Delon, J., and Bernot, M. Wasserstein barycenter and its application to texture mixing. In Scale Space and Variational Methods in Computer Vision, pp.\ 435--446, 2012

    2012

  31. [39]

    Optimal transport for applied mathematicians

    Santambrogio, F. Optimal transport for applied mathematicians. Birk \"a user, NY , 55 0 (58-63): 0 94, 2015

    2015

  32. [40]

    Smooth strongly convex regression

    Simonetto, A. Smooth strongly convex regression. In 2020 28th European Signal Processing Conference, pp.\ 2130--2134, 2021

    2020

  33. [41]

    Convolutional W asserstein distances: Efficient optimal transportation on geometric domains

    Solomon, J., de Goes, F., Peyr\' e , G., Cuturi, M., Butscher, A., Nguyen, A., Du, T., and Guibas, L. Convolutional W asserstein distances: Efficient optimal transportation on geometric domains. ACM Transactions on Graphs, 34 0 (4), 2015

    2015

  34. [42]

    and Panaretos, V

    Zemel, Y. and Panaretos, V. M. Fréchet means and Procrustes analysis in Wasserstein space . Bernoulli, 25 0 (2): 0 932 -- 976, 2019

    2019

  35. [43]

    Wasserstein autoregressive models for density time series

    Zhang, C., Kokoszka, P., and Petersen, A. Wasserstein autoregressive models for density time series. Journal of Time Series Analysis, 43 0 (1): 0 30--52, 2022

    2022

  36. [44]

    and M \"u ller, H.-G

    Zhu, C. and M \"u ller, H.-G. Geodesic optimal transport regression. arXiv preprint arXiv:2312.15376, 2023

    2023 arXiv

  37. [45]

    and Müller, H.-G

    Zhu, C. and Müller, H.-G. Autoregressive optimal transport models . Journal of the Royal Statistical Society Series B: Statistical Methodology, 85 0 (3): 0 1012--1033, 2023

    2023

  38. [46]

    Wasserstein k -means for clustering probability distributions

    Zhuang, Y., Chen, X., and Yang, Y. Wasserstein k -means for clustering probability distributions. In Proceedings of 36th Conference on Neural Information Processing Systems, 2022

    2022

  39. [47]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.