sinkhorn algorithm

This work addresses the data silo problem among clients in federated learning by adapting entropy-regularized optimal transport (Sinkhorn) to a distributed setting—its first such application—enabling collaborative discrete OT across multiple clients. We propose several federated Sinkhorn variants (synchronous/asynchronous, fully connected/star topologies), unified under a distributed marginal-constrained optimization framework. Each variant achieves privacy preservation and resource efficiency via local Sinkhorn iterations coupled with periodic potential function exchanges. We establish theoretical convergence guarantees for all variants and, for the first time, quantify the computational–communication trade-off in relation to problem scale and marginal sparsity. Extensive experiments on synthetic data and a real-world financial risk assessment task validate efficacy; empirical analysis characterizes the Pareto frontier between communication cost and accuracy across network topologies.

This paper addresses the challenge of achieving high-accuracy and efficient solutions to discrete optimal transport (OT) under weak regularization. We propose MDOT, a novel framework that constructs a sequence of dual problems via temperature annealing and mirror descent, and—crucially—introduces the first GPU-accelerated preconditioned nonlinear conjugate gradient method (PNCG) for solving them. Theoretically, we establish that Sinkhorn’s convergence rate can surpass existing non-asymptotic bounds. Methodologically, PNCG is pioneered for weakly regularized OT, markedly improving solution accuracy and numerical robustness. Empirically, on problems of size $n = 4096$, MDOT accelerates computation by several-fold over state-of-the-art solvers while consistently delivering high-accuracy feasible solutions. Complexity analysis confirms an empirical scaling of $ ilde{O}(n^2)$ at medium precision and $ ilde{O}(n^{5/2})$ at high precision.

This paper investigates the stability and exponential convergence of the Sinkhorn algorithm for entropy-regularized optimal transport. Focusing on quadratic cost, it establishes a Wasserstein stability theory based on semi-concavity assumptions, yielding the first global exponential convergence guarantee under log-concave marginals. It derives sharp convergence rate bounds with linear dependence on the regularization parameter. The analysis is extended to novel non-compact, unbounded settings—including Riemannian manifolds, elastic costs, and light-tailed marginals. Methodologically, the work integrates semi-concavity analysis, uniform upper bounds on the Hessian of Sinkhorn potentials, and refined Wasserstein distance estimation. Collectively, this provides the first unified exponential convergence guarantee for a broad class of generalized cost functions and marginal distributions. The derived rates improve upon prior results, significantly expanding the theoretical applicability of the Sinkhorn algorithm.

This paper investigates the bidirectional causal optimal transport (OT) problem with adapted structural coupling. To exploit its dynamic programming structure, we propose the first fitted value iteration (FVI) framework, employing deep neural networks to approximate the value function. Theoretically, under assumptions of concentrability and approximation completeness, we derive a sample complexity upper bound based on local Rademacher complexity and verify that suitably structured neural networks satisfy the required conditions. Experimentally, our method significantly outperforms linear programming and adapted Sinkhorn algorithms in computational efficiency as the time horizon increases, while maintaining controllable accuracy; it further exhibits strong scalability and practical feasibility. The core contribution lies in systematically introducing FVI to bidirectional causal OT—thereby establishing the first model-free approximate solution framework for this problem and filling a critical theoretical and methodological gap in the literature.

To address the high computational cost and curse-of-dimensionality challenges in sampling optimal transport (OT) couplings for large-scale, high-dimensional data, this paper proposes an efficient learning framework based on score-based generative models. Specifically, conditioned on source samples, it iteratively generates target samples following the Sinkhorn-regularized OT coupling via Langevin dynamics. Crucially, it jointly parameterizes the score function and Sinkhorn potential functions—enabling, for the first time, end-to-end co-learning of score-based generation and OT coupling. We theoretically establish the convergence of gradient descent on the network parameters under mild assumptions. Experiments demonstrate that our method significantly improves both accuracy and speed of coupling estimation across diverse large-scale OT tasks, while maintaining scalability and practical applicability.

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 computational challenges of large-scale entropy-regularized optimal transport, which is notoriously difficult to solve efficiently due to its high complexity. Existing GPU-based methods struggle to balance convergence speed and parallel efficiency. To overcome this limitation, we propose a high-performance GPU solver that synergistically combines the strengths of quasi-Newton methods and the Sinkhorn algorithm. Our approach introduces several key innovations: amortized sign analysis, asynchronous Sinkhorn iterations, and fused gradient kernels, complemented by sparse-plus-low-rank approximations, asynchronous computation, and optimized memory access patterns. While preserving theoretical convergence guarantees, the proposed method consistently outperforms state-of-the-art GPU solvers across multiple benchmark tasks, achieving substantial acceleration.

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 inefficiency of conventional distributed Sinkhorn algorithms in dynamic optimal transport, which suffer from frame-by-frame sequential processing and frequent synchronization. To overcome this, the authors propose TemporalSinkhorn, a temporally parallel executor that accelerates computation by batching future candidate solutions and incorporating a correction mechanism, all while preserving output accuracy guarantees. Key innovations include a centralized row-sharded certificate–based secure prefix scheme, an online auditing milestone guided by a forgetting rate, soft c-transform warm-starting, batched Sinkhorn updates, and posterior residual verification. Experiments on four A100 GPUs demonstrate speedups of 1.15–1.47× over sequential auditing, outperforming serial warm-starting by 1.42–3.55×, and achieving 3.05–3.63× acceleration in Flow Matching tasks—all without any loss in accuracy.

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.