# Tukey's Depth for Object Data

arXiv: https://arxiv.org/abs/2109.00493 · HTML: https://mlsanity.com/paper/2109.00493/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Sep 2021.

## Built on by (most cited first, 4 of 4)

- [Statistical Depth for Ranking and Characterizing Transformer-Based Text Embeddings](https://mlsanity.com/paper/2310.15010/) (2023)
- [Robust Estimation in metric spaces: Achieving Exponential Concentration with a Fr'echet Median](https://mlsanity.com/paper/2504.14161/) (2025)
- [Breakdown properties of optimal transport maps: general transportation costs](https://mlsanity.com/paper/2603.16005/) (2026)
- [A Trainable Centrality Framework for Modern Data](https://mlsanity.com/paper/2511.22959/) (2025)

## Abstract (arXiv)

We develop a novel exploratory tool for non-Euclidean object data based on data depth, extending the celebrated Tukey's depth for Euclidean data. The proposed metric halfspace depth, applicable to data objects in a general metric space, assigns to data points depth values that characterize the centrality of these points with respect to the distribution and provides an interpretable center-outward ranking. Desirable theoretical properties that generalize standard depth properties postulated for Euclidean data are established for the metric halfspace depth. The depth median, defined as the deepest point, is shown to have high robustness as a location descriptor both in theory and in simulation. We propose an efficient algorithm to approximate the metric halfspace depth and illustrate its ability to adapt to the intrinsic data geometry. The metric halfspace depth was applied to an Alzheimer's disease study, revealing group differences in the brain connectivity, modeled as covariance matrices, for subjects in different stages of dementia. Based on phylogenetic trees of 7 pathogenic parasites, our proposed metric halfspace depth was also used to construct a meaningful consensus estimate of the evolutionary history and to identify potential outlier trees.