Abstract:
Transport-based quantiles provide a canonical way to extend univariate quantiles to multivariate distributions through optimal transport. In this talk, we study the influence function of the transport-based quantile map $\mathbf{Q}_P$ , defined as the transport map pushing a fixed reference measure $\mu$ forward to a target distribution $P$. For Huber-type perturbations $P_t = (1 − t)P + t\delta_{x_0}$, we establish the existence and uniqueness of the first-order limit $$\mathbf{I}(x_0; \mathbf{Q}_P (z)) := \lim_{t\to 0}\frac{\mathbf{Q}(1−t)P +t\delta_{x_0}(z) −\mathbf{Q}_P (z)}{t}$$ away from the point $x_0 = \mathbf{Q}_P (z)$. We show that this influence function admits the representation $\mathbf{I}(x_0; \mathbf{Q}_P (z)) = \nabla G_{x_0}(z)$, where the potential $G_{x_0}$ is characterized by a uniformly elliptic equation with a Dirac source and Neumann boundary condition. Next, we show that, in dimension $d \geq 2$, the influence function has a pole-type singularity. More precisely, for fixed $z \in\text{int}(\Omega_\mu), the influence function remains bounded when the perturbation point $x_0$ is such that $\mathbf{F}_P (x_0)$ stays away from $z$, but it diverges as $x_0 \to \mathbf{Q}_P (z)$, equivalently as $\mathbf{F}_P (x_0) → z$. In fact, $\mathbf{I}(x_0; \mathbf{Q}_P (z)) \asymp \frac 1 {\|z−\mathbf{F}_P (x_0)\|^{d−1}}$, where $\mathbf{F}_P =\mathbf{Q}^{−1}_P$ is the transport-based distribution function. This singular behavior contrasts with the bounded influence function of univariate quantiles and implies that the influence-function random variable $\mathbf{I}(X; \mathbf{Q}_P (z))$, $X \sim P$, has infinite second moment. We also provide numerical evidence suggesting that empirical transport quantiles may exhibit stable-type, non-Gaussian fluctuations.