AI 中文总结
该研究通过最优传输解决了指定Spearman footrule时Spearman rho最大值的公开问题，得到二者的精确可达区域，并将其应用于有限排序、广义混合性及Chatterjee秩相关的界推导。
AI 中文摘要
我们解决了一个公开问题：当指定Spearman footrule时，确定Spearman rho的最大值，从而完整得到这两个量的精确可达区域。为证明该结果，我们将 underlying copula优化问题重新表述为带线性矩约束的最优传输问题，通过匹配可行的对偶势及其接触集构造唯一最优耦合。等价地，该耦合在所有满足指定均值𝔼|U-V|的U,V~𝒰(0,1)的耦合中，最小化|U-V|的方差。我们的主要结果有多项应用：第一，在有限排序语境中，我们得到Spearman footrule距离与相关二次秩差之间的改进柯西-施瓦茨不等式；第二，在广义混合性框架中，我们刻画了U',V'~𝒰(-1/2,1/2)时|U'+V'|的可达常数值，并确定给定均值下的最小二次偏差；第三，我们利用Chatterjee秩相关ξ(X,Y)（可检测Y对X的复杂函数依赖）与copula相关比（一种基于秩的解释方差分数）通过Spearman footrule和Spearman rho的条件独立同分布表示，推导了二者的显式界。
英文摘要
We solve the open problem of determining the maximal value of Spearman's rho when Spearman's footrule is prescribed, thereby completing the exact attainable region of these two quantities. To prove this result, we reformulate the underlying copula optimization problem as an optimal transport problem with a linear moment constraint and construct the unique optimal coupling through a matching feasible dual potential and its contact set. Equivalently, this coupling minimizes the variance of $|U-V|$ among all couplings of $U,V\sim\mathcal{U}(0,1)$ with prescribed mean $\mathbb{E}|U-V|$. Our main result admits several applications: First, in the context of finite rankings, we obtain an improved Cauchy--Schwarz inequality between Spearman's footrule distance and the associated quadratic rank difference. Second, in the framework of generalized mixability, we characterize the attainable constant values of $|U'+V'|$, for $U',V'\sim\mathcal{U}(-1/2,1/2)$, and determine the minimal quadratic deviation for a given mean. Third, we derive explicit bounds relating Chatterjee's rank correlation $ξ(X,Y)$, which can detect complex functional dependence of $Y$ on $X$, to the copula correlation ratio--a rank-based fraction of explained variance--by exploiting their conditional i.i.d. representations in terms of Spearman's footrule and Spearman's rho.
Comments46 pages, 7 figures
