The Influence Function of Transport-based Quantiles

Technical Summary: The Influence Function of Transport-based Quantiles

Problem Statement
Transport-based quantiles offer a canonical extension of univariate quantiles to multivariate distributions via optimal transport (OT). While recent literature has established the existence of high breakdown points for these estimators under suitable conditions, their robustness properties regarding infinitesimal perturbations—specifically the influence function (IF)—have not been systematically investigated. This paper addresses the gap in understanding the stability of the transport quantile map $Q_P$ (the OT map pushing a fixed reference measure $\mu$ to a target distribution $P$) when $P$ is subjected to Huber-type contamination, defined as $P_t = (1-t)P + t\delta_{x_0}$.

Methodology
The authors employ a rigorous analysis rooted in the partial differential equation (PDE) characterization of optimal transport maps. Unlike classical approaches that rely on the implicit function theorem, which fails here due to the non-smooth nature of the point-mass perturbation, the paper utilizes a multi-step analytical strategy:

1. $L^2$ Stability Estimates: The analysis begins by establishing $L^2$ bounds on the deviation of the perturbed transport map $Q_{P_t}$ from the unperturbed map $Q_P$. The authors prove that the squared $L^2(\mu)$ displacement scales as $O(t^{1+2/d})$ for $d \ge 3$ and $O(t^2 |\log t|)$ for $d=2$.
2. Linearization and $L^q$ Bounds: To analyze the first-order limit (the influence function), the authors study the sequence $\xi_t = (Q_{P_t} - Q_P)/t$. They demonstrate that while this sequence diverges in $L^2$, it remains uniformly bounded in $L^q(\Omega_\mu)$ for a specific range of $q \in (1, \frac{d(d+2)}{d^2+2d-4})$.
3. PDE Characterization: The limit of the potential function associated with $\xi_t$ is identified as the unique weak solution to a uniformly elliptic PDE with a Dirac source term and homogeneous Neumann boundary conditions.
4. Localization and Singularity Analysis: By excluding the perturbation point and utilizing local maximum principles (via Moser iteration) and Schauder estimates, the authors derive local $C^{2,\alpha}$ bounds away from the singularity. This allows for a precise characterization of the behavior near the singularity.

Key Contributions and Results

• Existence and Representation: The paper establishes the existence and uniqueness of the influence function $I(x_0; Q_P(z))$ for all $x_0 \neq Q_P(z)$. It is shown that $I(x_0; Q_P(z)) = \nabla G_{x_0}(z)$, where $G_{x_0}$ is the potential solving the elliptic PDE:
  $$\text{div}(f_\mu [\nabla Q_P]^{-1} \nabla G_{x_0}) = \mu - \delta_{z_0}$$
  where $z_0 = F_P(x_0)$ is the transport distribution function evaluated at the perturbation point, and $F_P = Q_P^{-1}$.

• Pole-Type Singularity: A central finding is that for dimensions $d \ge 2$, the influence function exhibits a pole-type singularity. Specifically, as the perturbation point $x_0$ approaches the quantile $Q_P(z)$ (equivalently, as $F_P(x_0) \to z$), the norm of the influence function diverges:
  $$\|I(x_0; Q_P(z))\| \asymp \frac{1}{\|z - F_P(x_0)\|^{d-1}}$$
  This contrasts sharply with the bounded influence function of univariate quantiles ($d=1$).

• Geometric Characterization of Preimages: As a byproduct of the proof strategy, the paper characterizes the set $K_{x_0, t} = Q_{P_t}^{-1}(x_0)$, which represents the points in the reference domain mapped to the perturbation location. For $d \ge 3$, this set behaves asymptotically as a ball of radius $t^{1/d}$ as $t \to 0$.

• Implications for Empirical Quantiles: The unbounded nature of the influence function implies that the random variable $I(X; Q_P(z))$ for $X \sim P$ has an infinite second moment ($I \notin L^2(P)$). Consequently, the empirical transport quantile $Q_{P_n}(z)$ does not admit a standard $\sqrt{n}$-asymptotically linear representation with a square-integrable influence function.

• Conjecture on Heavy-Tailed Fluctuations: Based on the tail behavior of the influence function (which decays as $\lambda^{-d/(d-1)}$), the authors conjecture that the empirical transport quantile converges at a non-parametric rate $r_n$ (where $r_n \sim n^{1/d}$ for $d \ge 3$) and follows a stable, non-Gaussian limiting distribution. Numerical experiments in dimension $d=3$ support this conjecture, showing that the empirical distribution aligns with the linear statistic derived from the influence function and exhibits heavy tails, deviating from Gaussian behavior even at large sample sizes.

Significance
The paper fundamentally alters the understanding of the robustness of transport-based quantiles. While these estimators possess high breakdown points (resistance to gross errors), the discovery of a pole-type singularity in their influence function reveals a distinct sensitivity to "inliers" (points close to the quantile of interest). This singular behavior explains why standard asymptotic normality fails and suggests that the limiting distribution of empirical transport quantiles is heavy-tailed and potentially stable. The work provides the necessary theoretical framework (via PDE characterization) to analyze these non-standard asymptotic behaviors, bridging optimal transport theory with robust statistics in the multivariate setting.