# Private Identity Testing for High-Dimensional Distributions

arXiv: https://arxiv.org/abs/1905.11947 · HTML: https://mlsanity.com/paper/1905.11947/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 May 2019.

## Built on by (most cited first, 15 of 15)

- [Robust and Differentially Private Mean Estimation](https://mlsanity.com/paper/2102.09159/) (2021)
- [Covariance-Aware Private Mean Estimation Without Private Covariance Estimation](https://mlsanity.com/paper/2106.13329/) (2021)
- [Private Robust Estimation by Stabilizing Convex Relaxations](https://mlsanity.com/paper/2112.03548/) (2021)
- [From Robustness to Privacy and Back](https://mlsanity.com/paper/2302.01855/) (2023)
- [Privately Learning Mixtures of Axis-Aligned Gaussians](https://mlsanity.com/paper/2106.02162/) (2021)
- [Private High-Dimensional Hypothesis Testing](https://mlsanity.com/paper/2203.01537/) (2022)
- [The Test of Tests: A Framework For Differentially Private Hypothesis Testing](https://mlsanity.com/paper/2302.04260/) (2023)
- [Federated Nonparametric Hypothesis Testing with Differential Privacy Constraints: Optimal Rates and Adaptive Tests](https://mlsanity.com/paper/2406.06749/) (2024)
- [Lower Bounds on the Total Variation Distance Between Mixtures of Two Gaussians](https://mlsanity.com/paper/2109.01064/) (2021)
- [Sample-Optimal Locally Private Hypothesis Selection and the Provable Benefits of Interactivity](https://mlsanity.com/paper/2312.05645/) (2023)
- [Optimal Algorithms for Augmented Testing of Discrete Distributions](https://mlsanity.com/paper/2412.00974/) (2024)
- [Differentially Private Partial Set Cover with Applications to Facility Location](https://mlsanity.com/paper/2207.10240/) (2022)
- [The power of private likelihood-ratio tests for goodness-of-fit in frequency tables](https://mlsanity.com/paper/2109.09630/) (2021)
- [Nearly-Linear Time Private Hypothesis Selection with the Optimal Approximation Factor](https://mlsanity.com/paper/2506.01162/) (2025)
- [Differentially Private Verification of Distribution Properties](https://mlsanity.com/paper/2604.10819/) (2026)

## Abstract (arXiv)

In this work we present novel differentially private identity (goodness-of-fit) testers for natural and widely studied classes of multivariate product distributions: Gaussians in ℝd with known covariance and product distributions over ± 1d. Our testers have improved sample complexity compared to those derived from previous techniques, and are the first testers whose sample complexity matches the order-optimal minimax sample complexity of O(d1/2/α2) in many parameter regimes. We construct two types of testers, exhibiting tradeoffs between sample complexity and computational complexity. Finally, we provide a two-way reduction between testing a subclass of multivariate product distributions and testing univariate distributions, and thereby obtain upper and lower bounds for testing this subclass of product distributions.