# Multivariate density estimation from privatised data: universal consistency and minimax rates

arXiv: https://arxiv.org/abs/2107.12649 · HTML: https://mlsanity.com/paper/2107.12649/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jul 2021.

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

- [About the Cost of Central Privacy in Density Estimation](https://mlsanity.com/paper/2306.14535/) (2023)
- [On Rate-Optimal Partitioning Classification from Observable and from Privatised Data](https://mlsanity.com/paper/2312.14889/) (2023)
- [On public and private binary classification with metric space valued predictors](https://mlsanity.com/paper/2605.31184/) (2026)

## Abstract (arXiv)

We revisit the classical problem of nonparametric density estimation but impose local differential privacy constraints. Under such constraints, the original multivariate data X1,ldots,Xn ∈ ℝd cannot be directly observed, and all estimators are functions of the randomised output of a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and in this work we propose to add Laplace distributed noise to a discretisation of the location of a vector Xi. Based on these randomised data, we design a novel estimator of the density function, which can be viewed as a privatised version of the well-studied histogram density estimator. Our theoretical results include universal pointwise consistency and strong universal L1-consistency. In addition, a convergence rate over classes of Lipschitz functions is derived, which is complemented by a matching minimax lower bound. We illustrate the trade-off between data utility and privacy by means of a small simulation study.