# Differentially private projection-depth-based medians

arXiv: https://arxiv.org/abs/2312.07792 · HTML: https://mlsanity.com/paper/2312.07792/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 37 references that are in the map.
- First posted on arXiv 12 Dec 2023.

## Builds on (most cited first, 13 of 13)

- [Privately Learning High-Dimensional Distributions](https://mlsanity.com/paper/1805.00216/) (2018)
- [CoinPress: Practical Private Mean and Covariance Estimation](https://mlsanity.com/paper/2006.06618/) (2020)
- [Robust and Differentially Private Mean Estimation](https://mlsanity.com/paper/2102.09159/) (2021)
- [Differential privacy and robust statistics in high dimensions](https://mlsanity.com/paper/2111.06578/) (2021)
- [Differentially Private Assouad, Fano, and Le Cam](https://mlsanity.com/paper/2004.06830/) (2020)
- [Efficient Mean Estimation with Pure Differential Privacy via a Sum-of-Squares Exponential Mechanism](https://mlsanity.com/paper/2111.12981/) (2021)
- [Covariance-Aware Private Mean Estimation Without Private Covariance Estimation](https://mlsanity.com/paper/2106.13329/) (2021)
- [On the Sample Complexity of Privately Learning Unbounded High-Dimensional Gaussians](https://mlsanity.com/paper/2010.09929/) (2020)
- [Propose, Test, Release: Differentially private estimation with high probability](https://mlsanity.com/paper/2002.08774/) (2020)
- [Optimal Private Median Estimation under Minimal Distributional Assumptions](https://mlsanity.com/paper/2011.06202/) (2020)
- [Differentially private sub-Gaussian location estimators](https://mlsanity.com/paper/1906.11923/) (2019)
- [Exact Privacy Guarantees for Markov Chain Implementations of the Exponential Mechanism with Artificial Atoms](https://mlsanity.com/paper/2204.01132/) (2022)
- [Archimedes Meets Privacy: On Privately Estimating Quantiles in High Dimensions Under Minimal Assumptions](https://mlsanity.com/paper/2208.07438/) (2022)

## Built on by (most cited first, 3 of 3)

- [Tukey Depth Mechanisms for Practical Private Mean Estimation](https://mlsanity.com/paper/2502.18698/) (2025)
- [Improved subsample-and-aggregate via the private modified winsorized mean](https://mlsanity.com/paper/2501.14095/) (2025)
- [Computationally tractable robust differentially private mean estimation](https://mlsanity.com/paper/2606.12654/) (2026)

## Abstract (arXiv)

We develop (ε,δ)-differentially private projection-depth-based medians using the propose-test-release (PTR) and exponential mechanisms. Under general conditions on the input parameters and the population measure, (e.g. we do not assume any moment bounds), we quantify the probability the test in PTR fails, as well as the cost of privacy via finite sample deviation bounds. Next, we show that when some observations are contaminated, the private projection-depth-based median does not break down, provided its input location and scale estimators do not break down. We demonstrate our main results on the canonical projection-depth-based median, as well as on projection-depth-based medians derived from trimmed estimators. In the Gaussian setting, we show that the resulting deviation bound matches the known lower bound for private Gaussian mean estimation. In the Cauchy setting, we show that the ``outlier error amplification'' effect resulting from the heavy tails outweighs the cost of privacy. This result is then verified via numerical simulations. Additionally, we present results on general PTR mechanisms and a uniform concentration result on the projected spacings of order statistics, which may be of general interest.