# Elliptical Perturbations for Differential Privacy

arXiv: https://arxiv.org/abs/1905.09420 · HTML: https://mlsanity.com/paper/1905.09420/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2019.

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

- [Joint Privacy Enhancement and Quantization in Federated Learning](https://mlsanity.com/paper/2208.10888/) (2022)
- [Communication-Efficient and Privacy-Adaptable Mechanism for Federated Learning](https://mlsanity.com/paper/2501.12046/) (2025)
- [Pure Differential Privacy for Functional Summaries with a Laplace-like Process](https://mlsanity.com/paper/2309.00125/) (2023)

## Abstract (arXiv)

We study elliptical distributions in locally convex vector spaces, and determine conditions when they can or cannot be used to satisfy differential privacy (DP). A requisite condition for a sanitized statistical summary to satisfy DP is that the corresponding privacy mechanism must induce equivalent measures for all possible input databases. We show that elliptical distributions with the same dispersion operator, C, are equivalent if the difference of their means lies in the Cameron-Martin space of C. In the case of releasing finite-dimensional projections using elliptical perturbations, we show that the privacy parameter ep can be computed in terms of a one-dimensional maximization problem. We apply this result to consider multivariate Laplace, t, Gaussian, and K-norm noise. Surprisingly, we show that the multivariate Laplace noise does not achieve ep-DP in any dimension greater than one. Finally, we show that when the dimension of the space is infinite, no elliptical distribution can be used to give ep-DP; only (ε,δ)-DP is possible.