We introduce a novel neural network-based algorithm to compute optimal transport (OT) plans for general cost functionals. In contrast to common Euclidean costs, i.e., $\ell^1$ or $\ell^2$, such functionals provide more flexibility and allow using auxiliary information, such as class labels, to construct the required transport map. Existing methods for general costs are discrete and have limitations in practice, i.e. they do not provide an out-of-sample estimation. We address the challenge of designing a continuous OT approach for general costs that generalizes to new data points in high-dimensional spaces, such as images. Additionally, we provide the theoretical error analysis for our recovered transport plans. As an application, we construct a cost functional to map data distributions while preserving the class-wise structure.

翻译：我们提出了一种新颖的基于神经网络的算法，用于计算一般成本泛函下的最优传输方案。与常见的欧几里得成本（例如 $\ell^1$ 或 $\ell^2$ 范数）不同，此类泛函提供了更大的灵活性，并允许利用辅助信息（如类别标签）来构建所需的传输映射。现有处理一般成本的方法属于离散方法，在实践中存在局限性，例如无法提供样本外估计。我们致力于设计一种适用于一般成本的连续最优传输方法，以推广到高维空间（如图像）中的新数据点。此外，我们为所恢复的传输方案提供了理论误差分析。作为应用，我们构建了一个成本泛函，用于在映射数据分布的同时保持类间结构。