compute bures-wasserstein barycenter

This work addresses the challenge of computing conditional Fréchet means—i.e., Fréchet regression—on the Bures-Wasserstein manifold, which entails non-convex optimization lacking both theoretical guarantees and efficient algorithms. Focusing on the space of positive definite matrices, the authors establish sufficient conditions for the existence of conditional barycenters and prove that the associated objective function possesses no local maxima. Building on these theoretical insights, they propose a projection-free first-order Riemannian optimization algorithm with provable convergence and further extend it to a stochastic Riemannian framework suitable for large-scale settings. Empirical validation on real biological networks and large synthetic diffusion tensor imaging datasets demonstrates the method’s effectiveness and scalability.

Existing discrete Wasserstein barycenter algorithms suffer from poor scalability and require full-sample access. This paper proposes a novel modeling framework based on the Wasserstein gradient flow, reformulating barycenter computation as an energy minimization problem in the space of probability measures—naturally incorporating geometric structure and enabling explicit energy-based regularization. The method employs a minibatch sampling scheme, drastically reducing computational and memory overhead. Convergence is theoretically guaranteed via analysis leveraging the Polyak–Łojasiewicz inequality. Experiments on synthetic datasets and domain adaptation tasks demonstrate superior accuracy, efficiency, and robustness compared to state-of-the-art discrete methods and neural-network baselines. Key contributions include: (i) the first scalable gradient-flow paradigm for Wasserstein barycenters; (ii) interpretable, energy-driven regularization; and (iii) establishing a new standard for minibatch-based Wasserstein barycenter computation.

本文提出了一种投影黎曼梯度下降算法，解决了Bures-Wasserstein重心计算中维度独立的线性收敛问题，通过在单位步长下实现，改进了现有方法的效率。

Linear modeling of high-dimensional probability measures remains challenging due to the non-Euclidean geometry of the 2-Wasserstein space. Method: We propose the Linear Barycentric Coding Model (LBCM), grounded in the Linear Optimal Transport (LOT) metric, and integrate LOT with variational analysis and probability measure embedding to design efficient covariance estimation and missing-data imputation algorithms. Contribution/Results: We establish, for the first time, the closed-form solution of LBCM and prove its equivalence to the 2-Wasserstein barycenter under compatible measures. Theoretically, we show that LBCM achieves exact representation in one dimension, identify intrinsic bottlenecks to its high-dimensional generalization, and derive finite-sample computability guarantees and generalization error bounds. Empirically, LBCM demonstrates superior accuracy and efficiency on measure synthesis, covariance modeling, and data imputation tasks—providing the first framework for linearized analysis of probability measures with both theoretical rigor and practical applicability.

To address the sensitivity of the Wasserstein barycenter to outlier distributions, this paper proposes the Wasserstein median—defined as the Fréchet median under the 2-Wasserstein distance—as a robust alternative. Methodologically, we develop a generic iterative algorithmic framework built upon existing Wasserstein barycenter solvers and provide a rigorous proof of its convergence. Theoretically, we establish, for the first time, the existence, strong consistency, and outlier-robustness of the Wasserstein median. Empirically, experiments on synthetic and real-world data—including single-cell gene expression profiles and image distributions—demonstrate substantial improvements in robustness: the median reduces sensitivity to outliers by 40%–65% compared to the barycenter, while preserving interpretability and computational tractability. This work introduces a new paradigm and practical tool for robust summarization of collections of probability distributions.

This study addresses the computational difficulty and inefficiency of approximating universal barycenters for probability distributions with arbitrary weights on the Wasserstein simplex. To overcome this limitation, we propose BaryFM, a flow matching model that, for the first time, parameterizes the entire barycenter family across the Wasserstein simplex using a single network. By integrating an ordinary differential equation (ODE) solver, BaryFM efficiently transports marginal distributions to barycenters corresponding to arbitrary weights, thereby transcending the constraints of traditional fixed-weight computation and enabling continuous weight sampling. Extensive evaluations across ten benchmarks in four tasks, including domain adaptation, demonstrate that BaryFM achieves a superior average rank compared to fifteen baseline methods. Furthermore, its performance is comparable to that of specialized, non-universal solvers, highlighting its effectiveness as a general-purpose framework for Wasserstein barycenter computation.

This work addresses the challenge of defining weighted averages (barycenters) of probability measures on graph structures, where classical optimal transport suffers from geometric degeneracy. By leveraging the Riemannian geometry of the probability simplex induced by dynamic optimal transport, the authors propose an intrinsic gradient descent method to compute barycenters on graphs. This approach approximates the exponential map and its inverse through numerical geodesic approximation and discretization of the continuity equation. In the analysis phase, barycentric coordinates are obtained by solving a quadratic program based on geodesic distances. Experimental results demonstrate that the proposed method significantly outperforms static optimal transport approaches relying on graph distances or entropic regularization, thereby validating the effectiveness and superiority of the introduced intrinsic geometric framework for measure synthesis and analysis on graphs.

This work addresses the recovery of latent geometric structure from nonlinear observations in the space of probability measures. It introduces, for the first time, the Wasserstein Mahalanobis distance—a geometry-aware metric between distributions constructed by combining optimal transport displacement fields with covariance operators on tangent spaces. This approach establishes a covariance-adaptive geometric framework for distribution-valued data, which exactly recovers the Mahalanobis distance of latent variables under affine transformations and achieves controllable error under general smooth transformations. Both theoretical analysis and numerical experiments demonstrate that the proposed distance effectively reconstructs the underlying latent geometry in the context of nonlinear independent component analysis.

This work addresses the sensitivity of the classical Wasserstein barycenter to outliers and its insufficient robustness. The authors propose a displacement-level Huberization approach that incorporates the Huber loss into the optimal transport cost, yielding a robust barycenter that exhibits locally quadratic behavior while suppressing large displacements. This formulation naturally interpolates between the Wasserstein mean and median and achieves a breakdown point as high as 1/2. Theoretical analysis establishes the existence, stability, and asymptotic distribution of the proposed estimator. Numerical experiments corroborate its strong robustness against contaminated data and confirm its interpolation properties between mean- and median-type aggregation.

This work addresses the sensitivity of classical Wasserstein barycenters to outliers and their reliance on finite-moment assumptions, which limits their applicability to heavy-tailed or contaminated data. The authors introduce robust optimal transport into Wasserstein barycenter computation, proposing the Robust Wasserstein Barycenter (RWB). By integrating truncation and regularization techniques, RWB yields an estimator that is insensitive to outliers. The paper establishes theoretical guarantees for the existence and statistical consistency of RWB under mild conditions. Empirical evaluations on synthetic data, image processing tasks, and financial time series demonstrate that RWB significantly outperforms conventional methods, offering enhanced robustness and practical utility in real-world scenarios involving data corruption or heavy-tailed distributions.