optimal transport interpolation

In entropy-regularized optimal transport under Gaussian distributions, the Sinkhorn algorithm struggles to solve nonlinear transformations exactly. Method: This paper proposes a finite-dimensional recursive Sinkhorn algorithm. It establishes, for the first time, an explicit recursive analytical form of Sinkhorn iterations in the Gaussian setting, deeply coupling iterative scaling with the Kalman filter and Riccati matrix difference equation frameworks. Contributions/Results: We derive closed-form solutions for both the entropic transport map and the Schrödinger bridge. Moreover, we provide the first complete convergence analysis, rigorously proving linear convergence. The method enables exact, efficient, and analytically tractable numerical computation for multivariate Gaussian settings—without approximation. By unifying optimal transport, Schrödinger bridges, and filtering theory, it offers a novel tool for probabilistic modeling and dynamic inference.

This work addresses the high computational complexity and poor scalability of classical optimal transport (OT). To this end, we propose an efficient probabilistic measure analysis framework based on sliced optimal transport (SOT). Methodologically, we integrate integral geometry with statistical estimation: we introduce nonlinear projections and adaptive weighted slicing strategies to improve Monte Carlo approximation, and extend SOT to unbalanced, multi-marginal, and Gromov–Wasserstein settings. Theoretically and algorithmically, our framework unifies support for sliced Wasserstein distance computation, barycenter estimation, kernel construction, and embedding learning. Experiments demonstrate that the proposed approach preserves the rich geometric structure of OT while achieving significant scalability improvements. Its efficacy and practicality are validated across diverse machine learning, computer vision, and graphics tasks.

Optimal transport (OT) is a central framework for modeling distribution shifts. Because OT compares distributions directly in input space, a well-designed ground metric between observations is essential to ensure that the optimizer does not violate the true geometry of change. We propose Displacement-Reshaped Optimal Transport (ReshapeOT), a method that reshapes the ground metric by integrating observed sample displacements as an additional source of knowledge. Technically, ReshapeOT replaces the Euclidean metric with a Mahalanobis distance estimated from displacement second moments. This effectively carves expressways through the input space, inviting transport solutions that better align with observed displacements. Our method is computationally lightweight, integrates seamlessly into any OT solver that operates on a cost matrix, and can be kernelized for further flexibility. Experiments on synthetic and real-world data show that ReshapeOT achieves substantial gains in transport reliability. We further demonstrate our method's usefulness in two practical use cases.

Optimal transport (OT) suffers from quadratic space complexity under large-scale data due to the Sinkhorn algorithm, hindering scalable computation of exact bijective Monge maps. Method: We propose the first linear-space, log-linear-time OT framework capable of computing exact bijective Monge mappings. Our approach leverages the inherent co-clustering structure embedded in low-rank OT factors, enabling a hierarchical multi-scale partitioning and iterative refinement scheme. At each level, it progressively approximates the globally optimal bijective coupling via low-rank approximations augmented with Monge structural priors. Contribution/Results: Our method achieves the first scalable, full-rank OT solution on datasets exceeding one million points. It guarantees strictly linear memory complexity while matching the matching accuracy of exact OT—thereby substantially surpassing current scalability limits.

This work addresses the lack of a unified theoretical foundation for parameter identifiability, estimation consistency, and algorithmic convergence in feature-parameterized inverse optimal transport (IOT). It proposes an analytical framework based on Sinkhorn linearization and spectral surrogates, which—through the introduction of spectral sandwich inequalities, restricted Hessian analysis, and ℓ₁ regularization—establishes, for the first time, four core theoretical guarantees for IOT: global parameter identifiability, exact support recovery, a Lipschitz continuous inverse mapping, and monotonic convergence of gradient descent. The framework combines geometric transparency with spectral precision and further quantifies the Hölder stability of the projection map under model misspecification. Numerical experiments corroborate the theoretical findings.

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.

This work addresses the limitations of existing convergence bounds for the Sinkhorn–Knopp algorithm in the presence of outliers, which heavily depend on the regularization parameter or element-wise ratios and thus poorly reflect practical performance. To overcome this, we introduce the notion of “well-boundedness” to characterize the intrinsic quality of the dominant data structure and combine it with a pre-scaling technique to effectively isolate the influence of outliers. Building on this framework, we uncover a density-threshold-driven phase transition phenomenon in matrix scaling and establish a novel convergence analysis. Under the well-boundedness condition, the algorithm achieves ε-accuracy in only O(log(1/ε)) iterations, providing the first rigorous convergence guarantee that is independent of problem dimension, regularization cost, and outlier contamination.

This work addresses the challenge in partial optimal transport (POT) where only a fraction of mass needs to be transported due to mismatched marginal distributions or the presence of outliers. While existing Sinkhorn-based methods suffer from high computational complexity and limited scalability, we propose an accelerated Sinkhorn algorithm for POT (ASPOT) that, for the first time, integrates Nesterov-type acceleration into the Sinkhorn framework. By combining alternating minimization with entropy regularization, ASPOT reduces the computational complexity to $\mathcal{O}(n^{7/3} \varepsilon^{-5/3})$. Furthermore, we introduce an optimized strategy for selecting the entropy regularization parameter to enhance convergence rates. Both theoretical analysis and experiments on real-world scenarios demonstrate the superior efficiency and performance of ASPOT compared to existing approaches.

This work addresses the rigidity of classical optimal transport in scenarios where mass conservation does not hold and its poor sample complexity in high dimensions. The authors investigate the finite-sample theory of entropy-regularized unbalanced optimal transport, introducing a translation-invariant dual formulation and analyzing its geometric structure. They establish, for the first time, high-probability convergence bounds for empirical optimal couplings at the coupling level. Their analysis demonstrates that entropy regularization not only substantially mitigates the curse of dimensionality and reduces the required sample size but also enhances estimation stability, all while preserving compatibility with efficient solvers such as Sinkhorn algorithms.