We propose a novel amortized optimization method for predicting optimal transport (OT) plans across multiple pairs of measures by leveraging Kantorovich potentials derived from sliced OT. We introduce two amortization strategies: regression-based amortization (RA-OT) and objective-based amortization (OA-OT). In RA-OT, we formulate a functional regression model that treats Kantorovich potentials from the original OT problem as responses and those obtained from sliced OT as predictors, and estimate these models via least-squares methods. In OA-OT, we estimate the parameters of the functional model by optimizing the Kantorovich dual objective. In both approaches, the predicted OT plan is subsequently recovered from the estimated potentials. As amortized OT methods, both RA-OT and OA-OT enable efficient solutions to repeated OT problems across different measure pairs by reusing information learned from prior instances to rapidly approximate new solutions. Moreover, by exploiting the structure provided by sliced OT, the proposed models are more parsimonious, independent of specific structures of the measures, such as the number of atoms in the discrete case, while achieving high accuracy. We demonstrate the effectiveness of our approaches on tasks including MNIST digit transport, color transfer, supply-demand transportation on spherical data, and mini-batch OT conditional flow matching.

翻译：我们提出了一种新颖的分摊优化方法，通过利用从切片最优传输中导出的Kantorovich势能，预测多对测度间的最优传输计划。我们引入了两种分摊策略：基于回归的分摊（RA-OT）和基于目标的分摊（OA-OT）。在RA-OT中，我们构建了一个函数回归模型，将原始最优传输问题中的Kantorovich势能作为响应变量，将从切片最优传输中获得的势能作为预测变量，并采用最小二乘法估计这些模型。在OA-OT中，我们通过优化Kantorovich对偶目标函数来估计函数模型的参数。两种方法中，预测的最优传输计划均通过估计的势能恢复。作为分摊最优传输方法，RA-OT和OA-OT能够通过复用从先前实例中学习到的信息，快速近似求解不同测度对间的重复最优传输问题。此外，通过利用切片最优传输提供的结构，所提出的模型更加简洁，不依赖于测度的具体结构（例如离散情况下的原子数量），同时保持高精度。我们在MNIST数字传输、颜色迁移、球面数据的供需运输以及小批量最优传输条件流匹配等任务中验证了该方法的效果。