arXiv Machine Learning
Jun 4
Cone-Compatible Monge Geometry for High-Dimensional Ordered Optimal Transport
arXiv:2606. 04695v1 Announce Type: new Abstract: High-dimensional optimal transport is seldom available in closed form.
A Unified Kantorovich Duality for Multimarginal Optimal Transport
arXiv:2601. 17171v2 Announce Type: replace-cross Abstract: We study Kantorovich duality for multimarginal optimal transport (MOT) with bounded continuous cost functions.
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arXiv Machine Learning
Sep 22
Toscani-Fourier Distance on Probability Measures: Wasserstein Control, Topological Equivalence on Model Classes, and Duality
arXiv:2609.23163v1 Announce Type: cross Abstract: Comparing probability measures in machine learning trades transport geometry against computational cost: Wasserstein distances encode the geometry of...
arXiv Machine Learning
Jun 25
Structured Approximations of Measures
arXiv:2310. 09149v3 Announce Type: replace-cross Abstract: We study the approximation of probability measures in the Wasserstein-$p$ distance by structured classes of approximators, motivated by applications in imaging, machine learning, and physical measurement under sensor constraints.
arXiv Machine Learning
Aug 11
Constrained Learning with Universally Learnable Concept Classes
arXiv:2608. 08414v1 Announce Type: new Abstract: We study constrained statistical learning over infinite-dimensional hypothesis classes in the fully nonconvex setting, and establish universal PACC learnability of the solutions of dual algorithms: Probably Approximately Correct on Constraints, guaranteeing optimality and constraint satisfaction at once.
arXiv Statistics ML
Aug 27
Barycentric Weak Inner-Product Gromov-Wasserstein
The paper introduces a weak Gromov-Wasserstein (wGW) framework that compares source relations with relations between target conditional laws, focusing on inner-product relations and preserving conditional means. It defines the barycentric weak inner-product GW (wIGW) distance, proves existence of minimizers under finite second moments, and presents a ridge-regularized dual formulation leading to an iterative algorithm for finitely supported measures. Experiments on point clouds, graphs, and a PBMC multiome study demonstrate that mean-preserving target refinements can incur zero cost and improve atlas-based cell type transfer.
arXiv Machine Learning
Sep 23
Statistical Rates for Entropic Optimal Transport in the Discrete to SubGaussian Regime
arXiv:2609. 26647v1 Announce Type: cross Abstract: We study statistical rates in entropic optimal transport in the semi-discrete regime where one measure has finite support and the other is subGaussian.
arXiv Machine Learning
Sep 10
Stability of the Monge Map in Semi-Dual Optimal Transport
arXiv:2605.05569v5 Announce Type: replace-cross Abstract: This paper shows that the semi-dual formulation of the optimal transport problem has a degenerate saddle-point structure, and that its numeri...
arXiv Machine Learning
Jul 30
Persistence Spheres: a Bi-continuous Linear Representation of Measures for Partial Optimal Transport
arXiv:2603. 15384v2 Announce Type: replace-cross Abstract: We improve and extend persistence spheres, introduced in~\cite{pegoraro2025persistence}.
arXiv Machine Learning
Sep 15
Riemannian ascent--descent for nonconvex nonconcave minimax landscapes: convergence to basin saddle points and applications to distributionally robust optimization
The paper introduces a new convergence framework for solving distributionally robust optimization problems formulated as nonconvex, nonconcave minimax problems over a Euclidean space and a Riemannian manifold. It defines a "basin saddle point"—a locally defined Nash equilibrium—and proves that a Riemannian gradient ascent–descent algorithm converges to such points under a local Łojasiewicz growth condition. The authors apply this theory to a statistical risk DRO problem over Gaussian measures, deriving explicit convergence rates and constants in terms of data dimension, loss moments, and reference covariance.
arXiv Machine Learning
Sep 1
Uniform Statistical Convergence of Empirical Sinkhorn Potentials with Exponential and Polynomial Dependence on the Regularization Parameter
arXiv:2608.29152v1 Announce Type: cross Abstract: We study the empirical Sinkhorn estimator of the entropic optimal transport potentials under the uniform loss. Since the potentials are only unique u...
arXiv Machine Learning
Jun 9
Conditional Random Ordered Transport Spaces
arXiv:2606. 08113v1 Announce Type: new Abstract: A small Wasserstein distance does not certify that a transformation is admissible.
arXiv Machine Learning
Jul 14
The VC dimension of partial concept classes via Radon's theorem
arXiv:2607. 10751v1 Announce Type: new Abstract: Following Alon, Hanneke, Holzman, and Moran (FOCS 2021), we define a partial concept class (PCC) as a family of partial functions \(f: V\to\{0,1,\ast\}\); equivalently, its concepts partition the ground set into black ($f^{-1}(1)$), grey ($f^{-1}(\ast)$), and white parts ($f^{-1}(0)$).