The paper introduces Lifted PCA, a canonical extension of Euclidean PCA to the Wasserstein space of probability measures. By lifting variational formulations on the Stiefel and Grassmann manifolds, the authors derive equivalent minimization problems over probability measures that correspond to Wasserstein barycenters with generalized costs. The work establishes dual formulations, optimality conditions, connections to multimarginal optimal transport, statistical consistency, and computational algorithms, and demonstrates practical performance on distributional datasets.
Principal Component Analysis (PCA) is a fundamental tool for statistical dimension reduction, yet no canonical extension to distribution-valued data has emerged within the framework of optimal transpo...
arXiv:2606. 17196v1 Announce Type: cross Abstract: This paper is concerned with learning principal variations of random probability measures on $\mathbb{R}^m$ under the Wasserstein geometry.
arXiv:2405. 15768v2 Announce Type: replace-cross Abstract: In this paper, we address the classification of instances represented by distributions on a vector space rather than single points.
The paper introduces LOT Wassmap, an algorithm designed to uncover low‑dimensional structures in the Wasserstein space without computing a full pairwise distance matrix. It leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to accelerate computations, and provides theoretical guarantees on embedding quality even when only finite samples are available. Experiments show that LOT Wassmap achieves accurate embeddings that improve with larger sample sizes while significantly reducing computational cost compared to traditional pairwise‑distance‑based methods.
arXiv:2605. 09916v2 Announce Type: replace-cross Abstract: We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal transport in large-scale, non-Euclidean datasets.
The paper introduces a geometric framework for measuring how far empirical datasets deviate from the Gaussian family using optimal transport theory. It defines two new quantities—the relative Wasserstein angle and the orthogonal projection distance—based on the cone structure of the relative translation invariant quadratic Wasserstein space, and shows that the usual moment‑matching Gaussian is not generally the $W_2$‑nearest Gaussian. Closed‑form expressions are derived for one‑dimensional and several location–scale families, while a numerical approximation is proposed for higher dimensions, with experiments demonstrating convergence, stability, and the angle’s robustness as a non‑Gaussianity indicator.
arXiv:2606. 04695v1 Announce Type: new Abstract: High-dimensional optimal transport is seldom available in closed form.
arXiv:2606. 02047v1 Announce Type: cross Abstract: We introduce Convex Distance Operator Transport (CDOT), the first convex optimal transport framework that aligns distributions across heterogeneous domains by jointly preserving feature correspondence and intrinsic geometric structure.
The paper develops a formal Riemannian framework for the Sinkhorn divergence, a debiased entropic optimal transport metric. It proves Hadamard differentiability of the Sinkhorn metric tensor, derives the Levi‑Civita covariant derivative, and obtains geodesic and parallel transport equations as ODEs on a fixed RKHS. Using Banach‑space ODE theory, the authors linearize these equations to establish √ n‑consistent Gaussian limits for plug‑in estimators of the exponential map, logarithmic map, parallel transport, and the induced Sinkhorn distance, and show bootstrap consistency for all these objects.
arXiv:2606. 27767v1 Announce Type: new Abstract: Optimizing functionals over the space of probability measures is now ubiquitous in machine learning.
arXiv:2610. 36581v1 Announce Type: cross Abstract: Dimension-reduction methods such as PCA identify a subspace capturing most of the variation in the data, but not a distinguished basis within it.
arXiv:2603.07014v2 Announce Type: replace-cross Abstract: Regression with distribution-valued responses and Euclidean predictors has gained increasing scientific relevance. While methodology for univ...