# Improved Differentially Private Euclidean Distance Approximation

arXiv: https://arxiv.org/abs/2203.11561 · HTML: https://mlsanity.com/paper/2203.11561/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Mar 2022.

## Built on by (most cited first, 5 of 5)

- [Private Query Release via the Johnson-Lindenstrauss Transform](https://mlsanity.com/paper/2208.07410/) (2022)
- [Differential Privacy with Random Projections and Sign Random Projections](https://mlsanity.com/paper/2306.01751/) (2023)
- [Differentially Private One Permutation Hashing and Bin-wise Consistent Weighted Sampling](https://mlsanity.com/paper/2306.07674/) (2023)
- [Perturb-and-Project: Differentially Private Similarities and Marginals](https://mlsanity.com/paper/2406.04868/) (2024)
- [Distributed Differentially Private Data Analytics via Secure Sketching](https://mlsanity.com/paper/2412.00497/) (2024)

## Abstract (arXiv)

This work shows how to privately and more accurately estimate Euclidean distance between pairs of vectors. Input vectors x and y are mapped to differentially private sketches x' and y', from which one can estimate the distance between x and y. Our estimator relies on the Sparser Johnson-Lindenstrauss constructions by Kane & Nelson (Journal of the ACM 2014), which for any 0<α,β<1/2 have optimal output dimension k=Theta(α-2log(1/β)) and sparsity s=O(α-1log(1/β)). We combine the constructions of Kane & Nelson with either the Laplace or the Gaussian mechanism from the differential privacy literature, depending on the privacy parameters varepsilon and δ. We also suggest a differentially private version of Fast Johnson-Lindenstrauss Transform (FJLT) by Ailon & Chazelle (SIAM Journal of Computing 2009) which offers a tradeoff in speed for variance for certain parameters. We answer an open question by Kenthapadi et al.~(Journal of Privacy and Confidentiality 2013) by analyzing the privacy and utility guarantees of an estimator for Euclidean distance, relying on Laplacian rather than Gaussian noise. We prove that the Laplace mechanism yields lower variance than the Gaussian mechanism whenever δ<βO(1/α). Thus, our work poses an improvement over the work of Kenthapadi et al.~by giving a more efficient estimator with lower variance for sufficiently small δ. Our sketch also achieves pure differential privacy as a neat side-effect of the Laplace mechanism rather than the approximate differential privacy guarantee of the Gaussian mechanism, which may not be sufficiently strong for some settings.