optimal transport alignment

Frequent computation of optimal transport (OT) between multiple source probability distributions incurs prohibitive computational overhead due to repeated cost matrix construction and dense transport plan estimation. Method: We propose Anchor Space Optimal Transport (ASOT), which constrains transport to a low-dimensional, learnable anchor space—thereby avoiding redundant cost matrix assembly and pruning spurious transport paths. ASOT is the first unified approximation framework for multiple OT problems, with a theoretically derived upper bound on the 1-Wasserstein distance error. We design three anchor space learning strategies supporting variable-length distributions and GPU-accelerated parallelization. Results: Experiments on graph and image data demonstrate up to two orders-of-magnitude speedup over standard OT solvers, while maintaining high-fidelity distribution alignment. ASOT significantly reduces the computational complexity of solving multiple OT problems, with bounded and controllable approximation error.

In network alignment, embedding-based methods are sensitive to graph noise, while optimal transport (OT)-based approaches rely on handcrafted cost functions, hindering end-to-end training and generalization. To address these limitations, we propose the first jointly optimized, alternating co-enhancement framework that unifies embedding learning and optimal transport. Specifically, OT solutions guide robust embedding sampling, while dynamically learned graph neural network embeddings adaptively refine the OT cost function. The framework is fully differentiable and supports end-to-end training. On real-world cross-network benchmarks, it achieves a 16% improvement in mean reciprocal rank (MRR) and accelerates inference by 20× over state-of-the-art methods. Crucially, it simultaneously enhances noise robustness, generalizability, and computational efficiency—resolving longstanding trade-offs in the literature.

This work addresses the challenge of low-rank optimal transport (OT), which, despite its ability to reveal latent data structures and enhance statistical stability, is notoriously non-convex and NP-hard. The authors propose the first reduction of this problem to a clustering task, introducing a novel "transport clustering" algorithm: it first computes a full-rank OT solution to obtain correspondences and then clusters these to construct a low-rank transport plan. The method achieves a constant-factor approximation in polynomial time and provides theoretical approximation guarantees under negative-type metrics and kernel-based costs. Empirical evaluations demonstrate that the algorithm significantly outperforms existing low-rank OT solvers on both synthetic and large-scale high-dimensional datasets, offering a favorable combination of computational efficiency and theoretical rigor.

This paper addresses the Hierarchical Optimal Transport (HOT) problem under multi-source heterogeneous transportation costs—a computationally challenging generalization of standard optimal transport. Method: We propose an efficient solution framework grounded in chordal graph theory: (i) we introduce a novel algebraic characterization of hierarchical transport structure using chordal graphs, enabling reduction of HOT to a standard optimal transport problem; (ii) we design an algebraic composition method for cost matrices to model adversarial multi-choice transportation costs; and (iii) we develop the first polynomial-time algorithm for exact and efficient synthesis of hierarchical transport plans. Results: Experiments demonstrate that our approach significantly outperforms naive solvers in both computational efficiency and solution quality. It exhibits strong practicality and scalability in large-scale hierarchical transportation settings, offering a theoretically grounded and empirically robust alternative for structured optimal transport.

Optimal transport alignment tasks are often hindered by sparse and costly supervisory signals. To address this limitation, this work proposes AvAtar—the first framework to integrate active learning into optimal transport alignment. AvAtar leverages the adjoint method to efficiently compute the gradient influence of candidate samples on the global alignment objective and introduces a general-purpose utility function adaptable across diverse alignment tasks. By combining entropy-regularized optimal transport, conjugate gradient optimization, and gradient propagation analysis, the proposed approach substantially outperforms existing methods across three representative alignment benchmarks, demonstrating strong effectiveness, scalability, and generalization capability.

This work addresses the computational intractability of high-dimensional optimal transport couplings, which often leads existing flow-matching methods to suffer from either bias or excessive computational cost. The authors propose a novel approach that treats the prior distribution as a designable variable, constructing it via low-frequency image projection to simultaneously satisfy the optimal transport identity coupling and enable efficient sampling. This reformulation reduces the generative task to synthesizing high-frequency details alone. Notably, the method achieves efficient single-step generation without modifying the underlying flow model and further enhances sample quality through Gaussian interpolation. Experimental results demonstrate a more than two-fold reduction in trajectory curvature across all benchmarks, substantially improving performance in few-step and even single-step generation settings.

This work addresses the limitations of conventional low-rank optimal transport methods, which rely on first-order gradient updates requiring careful hyperparameter tuning and neglect the intrinsic geometric structure of the underlying optimization problem, thereby compromising efficiency and stability. The authors propose the first unified Riemannian optimization framework that models both balanced and unbalanced low-rank couplings as smooth submanifolds within the positive orthant, equipped with a Fisher–Rao product metric. This formulation yields closed-form expressions for Riemannian projections, retractions, and Hessian-vector products, eliminating the need for inner iterative loops. The approach seamlessly accommodates linear optimal transport, Gromov–Wasserstein alignment, and their unbalanced variants, while providing a rank-sufficiency certificate for global optimality. Experiments demonstrate that the proposed first- and second-order solvers consistently outperform existing methods across problems of varying scales, with per-iteration complexity scaling only linearly in the data size.

Existing methods for evaluating agent trajectories rely on binary success flags or exact matches to reference trajectories, making them unable to distinguish between reasonable solutions and lucky successes and highly sensitive to step reordering and granularity differences. This work proposes OTAP, a pseudo-metric that formulates trajectory evaluation as a structure-aware distance between an execution graph and a set of valid solution graphs. Built upon attribute dependency graphs, OTAP leverages unbalanced fused Gromov–Wasserstein optimal transport combined with soft-coupling matching. It exhibits dependency-preserving permutation invariance, bounded sensitivity to redundant steps, and naturally accommodates missing actions, hallucinations, and granularity mismatches. Experiments demonstrate that OTAP significantly differentiates valid from invalid trajectories under controlled perturbations and across three benchmarks, substantially outperforming semantic metrics and achieving optimal performance when the underlying dependency graph is accurately recovered.

In large-scale learning, minibatch optimal transport (OT) is commonly employed to approximate exact OT, yet the global coupling mechanism underlying this approximation remains unclear. This work formally defines the expected minibatch OT plan—as the expectation of independent empirical minibatch OT plans—and establishes its consistency and convergence rates, with a refined characterization of bias and convergence behavior in the semi-discrete setting. The proposed framework is applied to flow matching, yielding velocity fields with favorable regularity that guarantee a unique flow transporting a continuous source distribution to a discrete target distribution. Experiments on diatomic models, synthetic data, and image tasks validate the theoretical findings, revealing a trade-off between batch size and numerical integration error, and demonstrating significantly enhanced stability and efficiency in flow matching.