# On the breakdown point of transport-based quantiles

arXiv: https://arxiv.org/abs/2410.16554 · HTML: https://mlsanity.com/paper/2410.16554/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Oct 2024.

## Built on by (most cited first, 2 of 2)

- [Breakdown properties of optimal transport maps: general transportation costs](https://mlsanity.com/paper/2603.16005/) (2026)
- [Huber-Wasserstein barycenters for robust distribution-valued data](https://mlsanity.com/paper/2608.13131/) (2026)

## Abstract (arXiv)

Recent work has used optimal transport ideas to generalize the notion of (center-outward) quantiles to dimension d≥ 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∈ [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.