# Differentially private depth functions and their associated medians

arXiv: https://arxiv.org/abs/2101.02800 · HTML: https://mlsanity.com/paper/2101.02800/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jan 2021.

## Built on by (most cited first, 7 of 7)

- [Robustness Implies Privacy in Statistical Estimation](https://mlsanity.com/paper/2212.05015/) (2022)
- [Covariance-Aware Private Mean Estimation Without Private Covariance Estimation](https://mlsanity.com/paper/2106.13329/) (2021)
- [Easy Differentially Private Linear Regression](https://mlsanity.com/paper/2208.07353/) (2022)
- [A Polynomial Time, Pure Differentially Private Estimator for Binary Product Distributions](https://mlsanity.com/paper/2304.06787/) (2023)
- [Differentially Private Estimation via Statistical Depth](https://mlsanity.com/paper/2207.12602/) (2022)
- [Tukey Depth Mechanisms for Practical Private Mean Estimation](https://mlsanity.com/paper/2502.18698/) (2025)
- [Differentially Private Two-Stage Gradient Descent for Instrumental Variable Regression](https://mlsanity.com/paper/2509.22794/) (2025)

## Abstract (arXiv)

In this paper, we investigate the differentially private estimation of data depth functions and their associated medians. We introduce several methods for privatizing depth values at a fixed point, and show that for some depth functions, when the depth is computed at an out of sample point, privacy can be gained for free when nrightarrow ∞. We also present a method for privately estimating the vector of sample point depth values. Additionally, we introduce estimation methods for depth-based medians for both depth functions with low global sensitivity and depth functions with only highly probable, low local sensitivity. We provide a general result (Lemma 1) which can be used to prove consistency of an estimator produced by the exponential mechanism, provided the limiting cost function is sufficiently smooth at a unique minimizer. We also introduce a general algorithm to privately estimate a minimizer of a cost function which has, with high probability, low local sensitivity. This algorithm combines the propose-test-release algorithm with the exponential mechanism. An application of this algorithm to generate consistent estimates of the projection depth-based median is presented. Thus, for these private depth-based medians, we show that it is possible for privacy to be obtained for free when nrightarrow ∞.