geodesic optimal transport mapping

This work addresses the challenge of generative modeling on measure spaces, particularly the transport of measures over measures (metameasures), by introducing the Wasserstein-on-Wasserstein (WoW) framework. It extends flow matching to the Wasserstein space of probability measures for the first time, leveraging a nested optimal transport plan—comprising both outer and inner couplings—to naturally induce a velocity field and define a deterministic dynamical system. By integrating nested Wasserstein geometry, sliced Wasserstein distances, and linearized approximations, the method achieves substantially improved computational efficiency and numerical stability. The resulting generative trajectories are nearly straight in the Wasserstein sense, yielding state-of-the-art performance with strong theoretical coherence in point cloud and set generation tasks.

This work addresses the problem of efficiently learning Wasserstein geodesics and their associated optimal transport velocity fields directly from samples to model the dynamic transport process between a source and a target distribution. Building upon the dynamical formulation of optimal transport, the constrained optimization problem is reformulated as a minimax game, enabling joint approximation of the geodesic, optimal map, and full velocity field using deep neural networks. The proposed method constitutes the first purely sample-driven neural solver capable of handling general cost functions—including the quadratic cost—without requiring explicit density estimation. Experiments on both synthetic and real-world datasets demonstrate that the approach accurately reconstructs geodesics and velocity fields and enables direct sampling from the target distribution, thereby validating its effectiveness and broad applicability.

The Gromov–Wasserstein (GW) geometry lacks a systematic gradient flow theory and an intrinsic Riemannian structure. Method: We propose an implicit minimizing movement scheme under the inner-product-based GW distance (IGW), establishing the first IGW gradient flow framework with rigorous convergence guarantees. Theoretically, we prove convergence of the generalized minimizing movement scheme to a continuity equation, characterize its intrinsic Riemannian metric, derive a Benamou–Brenier-type energy functional, and formulate gradients via Otto calculus. Algorithmically, we design a mobility operator and an explicit velocity field construction, validated through Gaussian mixture model (GMM) representation and numerical interpolation. Results: Experiments demonstrate that IGW interpolation significantly outperforms classical Wasserstein interpolation, exhibiting superior robustness and geometric consistency—particularly in preserving global distributional structure and maintaining 2-Wasserstein alignment.

This work addresses the topological information loss and degraded geometric awareness inherent in the Sliced Wasserstein (SW) distance due to its reliance on linear projections. To resolve this, we propose the Tree-Sliced Wasserstein distance with Splitting Maps (TSW-SL)—the first extension of SW to metric tree structures. Our core innovation lies in introducing a splitting map and a tree-domain Radon transform, rigorously proving their invertibility and metric properties while preserving the closed-form optimal transport solution. Theoretically, this generalizes both the Radon transform and optimal transport frameworks to non-Euclidean, hierarchical domains. Algorithmically, TSW-SL enables efficient gradient flow optimization via differentiable tree projections. Experiments demonstrate that TSW-SL consistently outperforms SW and its variants in gradient flow simulation, image style transfer, and generative modeling—achieving superior geometric robustness without sacrificing computational efficiency.

This work addresses the challenge of geometrically consistent matching of topological features—particularly persistent homology cycles—across disparate data systems. We propose Topological Optimal Transport (TpOT), the first framework that deeply integrates optimal transport with persistent homology. TpOT constructs a measure-topological network and defines a differentiable, geometry-aware topological-geometric joint distance within its non-negatively curved geodesic metric space, leveraging hypergraph optimal transport, measure theory, and Riemannian-geometric optimization of transport plans. On point cloud data, TpOT significantly reduces topological distortion while producing geometrically plausible and interpretable cycle-level correspondences. Theoretically, we prove that the proposed distance satisfies all metric axioms. TpOT establishes the first differentiable matching paradigm for topological data analysis that simultaneously ensures geometric fidelity and topological faithfulness.

This work unifies the theoretical foundations of diffusion models and flow matching within the Riemannian geometric framework of Wasserstein space. It interprets diffusion processes as gradient flows of a free energy functional—formulated as an initial value problem—whereas flow matching corresponds to geodesics with fixed endpoints, cast as a boundary value problem. By leveraging Otto calculus, the Fokker–Planck equation, the Jordan–Kinderlehrer–Otto (JKO) scheme, and the Benamou–Brenier principle, the study rigorously characterizes both the equivalence and distinctions between these two paradigms. This geometric perspective places DDPM, DDIM, NCSN/SMLD, and Energy Matching within a common variational framework and elucidates how flow matching achieves efficient deterministic generation through Wasserstein geodesics.

This work systematically addresses statistical uncertainty in probability measures within the optimal transport framework. By extending classical optimal transport theory to the setting of random probability measures, it establishes—for the first time—the Riemannian geometric structure of the L²-type Wasserstein space, characterizing its metric and geodesics, and introduces a stochastic flow whose sample paths follow Wasserstein gradient flows. This approach unifies statistical inference and generative modeling, extends Schwartz’s posterior consistency theorem to the Wasserstein topology, and provides convergence guarantees for principled inference under random sampling. Furthermore, it offers a theoretical foundation compatible with stochastic token sampling in Transformer architectures.

This work addresses the challenge of efficiently converting probabilistic optimal transport couplings into deterministic maps on Riemannian manifolds to better suit learning tasks. To this end, the authors propose an intrinsic barycentric projection framework that defines an optimal deterministic representative via conditional Fréchet means under a geodesic squared loss, and introduces the Monge defect to quantify its deviation from the true Brenier–McCann map. They further develop a compatible log–exp local projection method in the tangent space that aligns with this deterministic mapping. Leveraging tools from Fréchet means, geodesic distances, and Riemannian gradient optimization, theoretical analysis and experiments on spherical data, SPD matrices, and EEG covariance manifolds demonstrate the variational optimality of the intrinsic projection and the local displacement efficacy of the tangent-space projection.

This work investigates the statistical limits and sample complexity lower bounds for estimating any valid transport map without relying on optimal transport (OT). Formalizing the estimation task within a minimax framework, the study integrates tools from optimal transport theory, minimax analysis, and stability-based hypothesis testing to establish, for the first time, rigorous statistical lower bounds for non-optimal yet valid transport maps. The results demonstrate that under standard stability conditions, the statistical difficulty of estimating an arbitrary valid transport map is comparable to that of estimating the OT map itself. However, when the stability assumption fails, alternative valid maps can substantially improve estimation accuracy, outperforming the OT-based approach.

This study addresses the limitation of existing flow matching methods that rely on Gaussian optimal transport, which struggle to handle complex distributions such as heavy-tailed ones. To overcome this, we propose Probabilistic Geodesic Flow Matching, extending the framework to location-scale families by defining geodesic paths on the manifold of probability distributions. By integrating neural ordinary differential equations (neural ODEs), our method enables optimal path planning within the probability space, effectively transcending the constraints of Euclidean geometry. This approach significantly enhances the capacity of generative models to capture non-uniform structures. Extensive experiments demonstrate that our method accurately models complex distributions across both synthetic and scientific datasets, achieving performance that surpasses or matches state-of-the-art models.