The Influence Function of Transport-based Quantiles

2026-07-21

AI Summary

This study investigates the robustness of multivariate quantiles defined via optimal transport under the Huber contamination model. By integrating optimal transport theory, elliptic partial differential equations with Dirac source terms and Neumann boundary conditions, and asymptotic analysis, the authors rigorously establish—for the first time—the existence of the influence function for high-dimensional transport quantiles and derive its explicit form. The influence function is shown to exhibit a pole-type singularity, with its magnitude diverging at a rate proportional to the distance raised to the power $-(d-1)$, thereby lacking a finite second moment and violating the classical bounded-influence assumption that holds for univariate quantiles. Numerical experiments further confirm that empirical transport quantiles display stable non-Gaussian fluctuation behavior.

Abstract

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.

Problem

• influence function
• transport-based quantiles
• optimal transport
• multivariate distributions
• Huber contamination

Innovation

• transport-based quantiles
• influence function
• optimal transport
• elliptic PDE
• multivariate robustness