On the breakdown point of transport-based quantiles

TL;DR

This work analyzes the robustness of transport-based multivariate quantiles defined via optimal transport maps. It establishes that the transport median has a breakdown point of $1/2$ and that transport-depth contours of order $\tau$ have breakdown point $\tau$, with a key link showing the breakdown point of a transport map equals the Tukey depth of the reference point. The results cover both finite-sample (discrete) and population (continuous) settings, and hold for natural choices of reference measures that maximize Tukey depth, such as halfspace-symmetric distributions with depth $1/2$. The findings provide theoretical guarantees for the robustness and computational tractability of transport-based quantiles and contours, guiding reference-measure selection and enabling reliable high-dimensional nonparametric inference.

Abstract

Recent work has used optimal transport ideas to generalize the notion of (center-outward) quantiles to dimension $d\geq 2$. We study the robustness properties of these transport-based quantiles by deriving their breakdown point, roughly, the smallest amount of contamination required to make these quantiles take arbitrarily aberrant values. We prove that the transport median defined in Chernozhukov et al.~(2017) and Hallin et al.~(2021) has breakdown point of $1/2$. Moreover, a point in the transport depth contour of order $τ\in [0,1/2]$ has breakdown point of $τ$. This shows that the multivariate transport depth shares the same breakdown properties as its univariate counterpart. Our proof relies on a general argument connecting the breakdown point of transport maps evaluated at a point to the Tukey depth of that point in the reference measure.

Figures (2)

• Figure 1: Example showing that the lower bound of \ref{['Theo:mainDiscrete']} is sharp. The reference and target datasets $u^{(n)}$ and $X^{(n)}$ are both $\{(0,0), (1,0), (0,1), (-1, 0), (0,-1)\}$. The Tukey depth of the origin is $3/5$ and the lower Tukey depth is $2/5$. Perturbing two points of $X^{(n)}$ as in the red dataset of the picture, we find that the optimal transport map from the blue the the red points maps $(0,0)$ to $(0,k)$, which escapes to the horizon. Therefore the breakdown point of the optimal transport map evaluated at the origin is $2/5$.
• Figure 2: Diagram illustrating the strategy for the proof of \ref{['Theo:mainCont']}. For the upper bound, we consider the mixture $(1-\varepsilon)P + \varepsilon Q_k$, where $Q_k$ is uniformly distributed on the ball $k v_u+\mathbb{B}$ for $v_u\in\mathop{\mathrm{arg\,min}}\limits_{v\in \mathcal{S}^{d-1}} \mu\left( \{z: \langle v, z - u \rangle \geq 0\} \right)$. If $\varepsilon > {\rm TD}(u;\mu)$ and a sequence $\{x_k\}_{k \in \mathbb{N}}$ with $x_k\in \partial \varphi_{\varepsilon,k}(u):=\partial \varphi_{\mu\to ( 1-\varepsilon ) P+\varepsilon Q_{k}}(u)$ remains bounded within a ball of radius $R$, then for sufficiently large $k$, the map $(\partial \varphi_{\varepsilon,k})^{-1}$ transforms the cone with vertex $x_k$ and angle $\theta_k$ into a set located to the right of the rays $r_1$ and $r_2$. As the ball escapes to the horizon, the angle $\theta_k$ tends to $0$, causing the image of the ball to be contained within the left halfspace defined by the hyperplane $H$, which is tangent to $v_u$. Consequently, as $\varepsilon > {\rm TD}(u;\mu)$, the $((1-\varepsilon)P+\varepsilon Q_k)$-measure of the cone exceeds ${\rm TD}(u;\mu)$, while its image under $(\partial \varphi_{\varepsilon,k})^{-1}$ converges to a set with $\mu$-probability of ${\rm TD}(u;\mu)$. This leads to a contradiction. The strategy for the lower bound follows a similar approach.

Theorems & Definitions (32)

• Theorem 1.2
• Corollary 1.3
• Remark 1.4
• Lemma 1.5
• Remark 1.6
• Remark 2.1
• Lemma 2.2
• proof
• Theorem 3.1
• Remark 3.2