Many machine learning problems involve data supported on curved spaces such as spheres, rotation groups, hyperbolic spaces, and general Riemannian manifolds, where Euclidean geometry can distort distances, averages, and the resulting optimal transport (OT) problem. Existing manifold OT methods have pursued amortized out-of-sample maps, while entropic regularization has made discrete OT more scalable, but these advantages have remained largely disjoint. We propose Entropic Riemannian Neural Optimal Transport (Entropic RNOT), a unified framework that combines intrinsic entropic OT with amortized out-of-sample evaluation on Riemannian manifolds. Our method learns a single target-side Schrödinger potential through a neural pullback parameterization, recovers the induced Gibbs coupling, and uses the resulting conditional laws to construct intrinsic transport surrogates. These include barycentric projections on Cartan-Hadamard manifolds and heat-smoothed conditional surrogates on stochastically complete manifolds, the latter turning possibly atomic target laws into absolutely continuous ones. For fixed regularization $\varepsilon>0$, we prove that the proposed hypothesis class recovers the entropic optimal coupling in strong probabilistic metrics. As consequences, barycentric surrogates converge in $L^2$, while heat-smoothed surrogates are stable at fixed heat time and asymptotically unbiased as the heat time vanishes. The guarantees hold for compactly supported data on possibly noncompact manifolds. Empirically, our method matches or improves over Euclidean, tangent-space, and log-Euclidean baselines on benchmarks over $\mathbb{S}^2$, $\mathrm{SO}(3)$, $\mathrm{SPD}(3)$, $\mathrm{SE}(3)$, and $\mathbb{H}^2$, scales favorably relative to discrete manifold Sinkhorn, and in a protein-ligand docking application, refines poses on $\mathrm{SE}(3)$ without retraining or per-instance optimization.

For the moment, it is worth noting that such sets of functions (e.g., Haar wavelets, Daubechies wavelets) are readily We are now in a position to present the main theorem of this subsection. To avoid repetition, we defer further discussions on the rates observed in Theorem A.1 to Remark 2.7 where a holistic In fact, by Proposition 1.1, there exists an optimal transport map Based on (B.2), the natural plug-in estimator of ρ Suppose that the same assumptions from Theorem 2.2 hold. B.2 Nonparametric independence testing: Optimal transport based Hilbert-Schmidt independence criterion Proposition B.2 shows that the test based on Further, when the sampling distribution is fixed, Proposition B.2 shows that In the following result (see Appendix C.2 for a proof), we show that if This section is devoted to proving our main results and is organized as follows: In Appendix C.1, we Further by Lemma D.2, we also have: ϕ Note that (C.10) immediately yields the following conclusions: S By (1.5) and some simple algebra, the following holds: null null null S Combining the above display with (C.9), we further have: null null null null 1 2 W Combining the above observation with Theorem 2.1, we have: lim sup For the next part, to simplify notation, let us begin with some notation. By using the exponential Markov's inequality coupled with the standard union Now by using [7, Theorem 2.10], we have P (B We are now in a position to complete the proof of Theorem 2.2 using steps I-III. Therefore, it is now enough to bound the right hand side of (C.17).

Optimal transport maps between two probability distributions $\mu$ and $\nu$ on $\R^d$ have found extensive applications in both machine learning and statistics. In practice, these maps need to be estimated from data sampled according to $\mu$ and $\nu$. Plug-in estimators are perhaps most popular in estimating transport maps in the field of computational optimal transport. In this paper, we provide a comprehensive analysis of the rates of convergences for general plug-in estimators defined via barycentric projections. Our main contribution is a new stability estimate for barycentric projections which proceeds under minimal smoothness assumptions and can be used to analyze general plug-in estimators. We illustrate the usefulness of this stability estimate by first providing rates of convergence for the natural discrete-discrete and semi-discrete estimators of optimal transport maps.

Abstract--Monotone gradient functions play a central role in solving the Monge formulation of the optimal transport (OT) problem, which arises in modern applications ranging from fluid dynamics to robot swarm control. When the transport cost is the squared Euclidean distance, Brenier's theorem guarantees that the unique optimal transport map satisfies a Monge-Amp ` ere equation and is the gradient of a convex function. In [1], [2], we proposed Monotone Gradient Networks (mGradNets), neural networks that directly parameterize the space of monotone gradient maps. In this work, we leverage mGradNets to directly learn the optimal transport mapping by minimizing a training loss function defined using the Monge-Amp ` ere equation. We empirically show that the structural bias of mGradNets facilitates the learning of optimal transport maps across both image morphing tasks and high-dimensional OT problems.

DISCRETE OPTIMAL TRANSPORT IS A STRONG AUDIO ADVERSARIAL A TT ACK A. Selitskiy, ABSTRACT In this paper, we show that discrete optimal transport (DOT) is an effective black-box adversarial attack against modern audio anti-spoofing countermeasures (CMs). Our attack operates as a post-processing, distribution-alignment step: frame-level WavLM embeddings of generated speech are aligned to an unpaired bona fide pool via entropic OT and a top-k barycentric projection, then decoded with a neural vocoder. Evaluated on ASVspoof2019 and ASVspoof5 with AASIST baselines, DOT yields consistently high equal error rate (EER) across datasets and remains competitive after CM fine-tuning, outperforming several conventional attacks in cross-dataset transfer. Ablation analysis highlights the practical impact of vocoder overlap. Results indicate that distribution-level alignment is a powerful and stable attack surface for deployed CMs.

The Wasserstein barycenter extends the Euclidean mean to the space of probability measures by minimizing the weighted sum of squared 2-Wasserstein distances. We develop a free-support algorithm for computing Wasserstein barycenters that avoids entropic regularization and instead follows the formal Riemannian geometry of Wasserstein space. In our approach, barycenter atoms evolve as particles advected by averaged optimal-transport displacements, with barycentric projections of optimal transport plans used in place of Monge maps when the latter do not exist. This yields a geometry-aware particle-flow update that preserves sharp features of the Wasserstein barycenter while remaining computationally tractable. We establish theoretical guarantees, including consistency of barycentric projections, monotone descent and convergence to stationary points, stability with respect to perturbations of the inputs, and resolution consistency as the number of atoms increases. Empirical studies on averaging probability distributions, Bayesian posterior aggregation, image prototypes and classification, and large-scale clustering demonstrate accuracy and scalability of the proposed particle-flow approach, positioning it as a principled alternative to both linear programming and regularized solvers.

For the moment, it is worth noting that such sets of functions (e.g., Haar wavelets, Daubechies wavelets) are readily We are now in a position to present the main theorem of this subsection. To avoid repetition, we defer further discussions on the rates observed in Theorem A.1 to Remark 2.7 where a holistic In fact, by Proposition 1.1, there exists an optimal transport map Based on (B.2), the natural plug-in estimator of ρ Suppose that the same assumptions from Theorem 2.2 hold. B.2 Nonparametric independence testing: Optimal transport based Hilbert-Schmidt independence criterion Proposition B.2 shows that the test based on Further, when the sampling distribution is fixed, Proposition B.2 shows that In the following result (see Appendix C.2 for a proof), we show that if This section is devoted to proving our main results and is organized as follows: In Appendix C.1, we Further by Lemma D.2, we also have: ϕ Note that (C.10) immediately yields the following conclusions: S By (1.5) and some simple algebra, the following holds: null null null S Combining the above display with (C.9), we further have: null null null null 1 2 W Combining the above observation with Theorem 2.1, we have: lim sup For the next part, to simplify notation, let us begin with some notation. By using the exponential Markov's inequality coupled with the standard union Now by using [7, Theorem 2.10], we have P (B We are now in a position to complete the proof of Theorem 2.2 using steps I-III. Therefore, it is now enough to bound the right hand side of (C.17).