从计划到映射：最优传输的非局部正则化
From plans to maps: Nonlocal regularization of optimal transport
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中文总结 AI 辅助
该研究提出最优传输的非局部正则化方法，连接Kantorovich与Monge形式，分析其极限行为并推导对偶形式与数值格式，实现两类问题的等价性及相关测度的存在性结果。
中文摘要 AI 辅助
我们引入最优传输的一种非局部正则化方法，其连接了Kantorovich与Monge两种形式。该正则化通过在对角线附近奇异的相互作用核来惩罚振荡。我们证明，对于足够强的奇异性，所有有限能量的传输计划均由一个映射诱导，从而使正则化后的Kantorovich问题与Monge问题等价。对于分数阶核，该正则项退化为分数阶Sobolev半范数，其允许跳间断，进而得到广泛类别的源测度与目标测度的存在性结果。我们推导了一阶最优性条件并分析了消失正则化极限：正则化泛函Γ收敛到经典Kantorovich问题，而一阶Γ极限在存在有限能量Monge极小化子（具有最小非局部能量）时会将其选出；当不存在此类极小化子时，我们确定了不同的渐近 regime，并在一个典型的质量拆分示例中建立了精确的标度律。最后，在将问题提升至传输计划的乘积空间后，我们利用共正函数锥推导了对偶形式，并提出了说明该正则化的不动点数值格式。
英文摘要
We introduce a nonlocal regularization of optimal transport that bridges Kantorovich and Monge formulations. The regularization penalizes oscillations through an interaction kernel singular near the diagonal. We show that, for sufficiently strong singularities, every finite-energy transport plan is induced by a map, yielding equivalence between the regularized Kantorovich and Monge problems. For fractional kernels, the regularizer reduces to a fractional Sobolev seminorm that can allow jump discontinuities, leading to existence results for broad classes of source and target measures. We derive first-order optimality conditions and analyze the vanishing-regularization limit. The regularized functionals $Γ$-converge to the classical Kantorovich problem, while the first-order $Γ$-limit selects, whenever they exist, finite-energy Monge minimizers with minimal nonlocal energy. When no such minimizer exists, we identify a different asymptotic regime and establish a sharp scaling law in a prototypical mass-splitting example. Finally, after lifting the problem to the product of transport plans, we derive a dual formulation using the cone of copositive functions and propose a fixed-point numerical scheme illustrating the regularization.
发表机构
- University of Twente（特温特大学）
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