AI 中文总结
研究二次型和扭曲最优传输中的菱形传输，在QOT框架下，对称一维边缘分布时菱形传输是相关QOT问题优化器，矩形成本时除边界情况是唯一极小值；在DOT框架中，菱形型传输是自然成本极小值，还确定了DOT与QOT的交集。
AI 中文摘要
菱形传输由菱形连接支撑上的均匀定律产生。由于经典最优传输（OT）目标在耦合中是仿射的，这种传输在经典设置中不是唯一的极小值。我们研究了更广泛的传输族，称为菱形型传输，在非经典设置中，如二次型最优传输（QOT）和扭曲最优传输（DOT），它们通常是非凸的。我们的主要结果在QOT框架内：对于对称的一维边缘分布，菱形传输是一大类成本依赖于坐标内距离的QOT问题的优化器。在DOT框架中，菱形型传输是一类自然成本的极小值，在特定例子中菱形传输是唯一的极小值。我们还确定了DOT和QOT的交集，它恰好对应于二次扭曲函数。
英文摘要
The diamond transport is generated by the uniform law on a diamond-shaped copula support. Since a classical optimal transport (OT) objective is affine in the coupling, this transport cannot be the unique minimizer in the classical setting. We study a broader family of transports, called diamond-type transports, in non-classical settings such as quadratic-form optimal transport (QOT) and distorted optimal transport (DOT), which are generally nonconvex. Our main results are within the QOT framework: for symmetric one-dimensional marginals, the diamond transport is an optimizer for a large class of QOT problems whose costs depend on within-coordinate distances. Examples include product costs under positive-definiteness and convexity conditions and, in particular, mixed rectangular costs. For rectangular costs, we show that the diamond transport is the unique minimizer except for boundary cases. In the DOT framework, diamond-type transports are minimizers for a natural class of cost, and the diamond transport is the unique minimizer in specialized examples. We also identify the intersection between DOT and QOT, which corresponds precisely to quadratic distortion functions.
Comments35 pages, 3 figures
