AI 中文总结
研究基于传输的分位数映射\(\mathbf{Q}_P\)的影响函数，通过对Huber污染的分析，证明其在特定条件下一阶极限存在并刻画了它，指出在高维有极点型奇点，与单变量分位数不同，还通过实验表明经验传输分位数有稳定型非高斯波动。
AI 中文摘要
基于传输的分位数通过最优传输将单变量分位数扩展到多元分布。本文研究了传输分位数映射\(\mathbf{Q}_P\)的影响函数，它被定义为将固定参考测度\(\mu\)向前推到目标分布\(P\)的最优传输映射。对于Huber污染\(P_t=(1-t)P+t\delta_{x_0}\)，证明了只要\(x_0\neq\mathbf{Q}_P(z)\)，一阶极限\(\mathbf{I}(x_0;\mathbf{Q}_P(z)):=\lim_{t\downarrow0}[\mathbf{Q}_{(1-t)P+t\delta_{x_0}}(z)-\mathbf{Q}_P(z)]/t\)存在，并对其进行了唯一刻画。具体来说，\(\mathbf{I}(x_0;\mathbf{Q}_P(z))=\nabla G_{x_0}(z)\)，其中\(G_{x_0}\)由一个具有狄拉克源和诺伊曼边界条件的一致椭圆方程刻画。在每个维度\(d\geq2\)中，这个影响函数具有极点型奇点。对于固定的\(z\in\operatorname{int}(\Omega_\mu)\)，当\(\mathbf{F}_P(x_0)\)远离\(z\)时它保持有界，但当\(x_0\to\mathbf{Q}_P(z)\)时发散，等价于当\(\mathbf{F}_P(x_0)\to z\)时。事实上，\(\|\mathbf{I}(x_0;\mathbf{Q}_P(z))\|\asymp\|z-\mathbf{F}_P(x_0)\|^{-(d - 1)}\)。这与单变量分位数的有界影响函数形成对比，并意味着对于\(X\sim P\)，\(\mathbf{I}(X;\mathbf{Q}_P(z))\)具有无限二阶矩。数值实验进一步表明经验传输分位数可能表现出稳定型非高斯波动。
英文摘要
Transport-based quantiles extend univariate quantiles to multivariate distributions via optimal transport. We study the influence function of the transport quantile map $\mathbf{Q}_P$, defined as the optimal transport map pushing a fixed reference measure $μ$ forward to a target distribution $P$. For the Huber contamination $P_t=(1-t)P+tδ_{x_0}$, we prove that the first-order limit $\mathbf{I}(x_0;\mathbf{Q}_P(z)) := \lim_{t\downarrow 0} [\mathbf{Q}_{(1-t)P+tδ_{x_0}}(z)-\mathbf{Q}_P(z)]/t$ exists whenever $x_0\ne \mathbf{Q}_P(z)$ and characterize it uniquely. Specifically, $\mathbf{I}(x_0;\mathbf{Q}_P(z))=\nabla G_{x_0}(z)$, where $G_{x_0}$ is characterized by a uniformly elliptic equation with a Dirac source and a Neumann boundary condition. In every dimension $d\ge 2$, this influence function has a pole-type singularity. For fixed $z\in\operatorname{int}(Ω_μ)$, it remains bounded when $\mathbf{F}_P(x_0)$ stays away from $z$, where $\mathbf{F}_P=\mathbf{Q}_P^{-1}$ is the transport-based distribution function, but diverges as $x_0\to\mathbf{Q}_P(z)$, equivalently as $\mathbf{F}_P(x_0)\to z$. In fact, $\|\mathbf{I}(x_0;\mathbf{Q}_P(z))\|\asymp\|z-\mathbf{F}_P(x_0)\|^{-(d-1)}$. This contrasts with the bounded influence function of univariate quantiles and implies that $\mathbf{I}(X;\mathbf{Q}_P(z))$, for $X\sim P$, has infinite second moment. Numerical experiments further suggest that empirical transport quantiles may exhibit stable-type non-Gaussian fluctuations.
