# Fast, Sample-Efficient, Affine-Invariant Private Mean and Covariance Estimation for Subgaussian Distributions

arXiv: https://arxiv.org/abs/2301.12250 · HTML: https://mlsanity.com/paper/2301.12250/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 27 references that are in the map.
- First posted on arXiv 28 Jan 2023.

## Builds on (most cited first, 16 of 16)

- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [The Discrete Gaussian for Differential Privacy](https://mlsanity.com/paper/2004.00010/) (2020)
- [Finite Sample Differentially Private Confidence Intervals](https://mlsanity.com/paper/1711.03908/) (2017)
- [Privately Learning High-Dimensional Distributions](https://mlsanity.com/paper/1805.00216/) (2018)
- [Differential privacy and robust statistics in high dimensions](https://mlsanity.com/paper/2111.06578/) (2021)
- [Robustness Implies Privacy in Statistical Estimation](https://mlsanity.com/paper/2212.05015/) (2022)
- [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)
- [Private and polynomial time algorithms for learning Gaussians and beyond](https://mlsanity.com/paper/2111.11320/) (2021)
- [Private Robust Estimation by Stabilizing Convex Relaxations](https://mlsanity.com/paper/2112.03548/) (2021)
- [FriendlyCore: Practical Differentially Private Aggregation](https://mlsanity.com/paper/2110.10132/) (2021)
- [A Private and Computationally-Efficient Estimator for Unbounded Gaussians](https://mlsanity.com/paper/2111.04609/) (2021)
- [New Lower Bounds for Private Estimation and a Generalized Fingerprinting Lemma](https://mlsanity.com/paper/2205.08532/) (2022)
- [A Primer on Private Statistics](https://mlsanity.com/paper/2005.00010/) (2020)
- [Privately Learning Subspaces](https://mlsanity.com/paper/2106.00001/) (2021)
- [A Fast Algorithm for Adaptive Private Mean Estimation](https://mlsanity.com/paper/2301.07078/) (2023)

## Built on by (most cited first, 22 of 22)

- [Better and Simpler Lower Bounds for Differentially Private Statistical Estimation](https://mlsanity.com/paper/2310.06289/) (2023)
- [Insufficient Statistics Perturbation: Stable Estimators for Private Least Squares](https://mlsanity.com/paper/2404.15409/) (2024)
- [PLAN: Variance-Aware Private Mean Estimation](https://mlsanity.com/paper/2306.08745/) (2023)
- [Federated Nonparametric Hypothesis Testing with Differential Privacy Constraints: Optimal Rates and Adaptive Tests](https://mlsanity.com/paper/2406.06749/) (2024)
- [Dimension-free Private Mean Estimation for Anisotropic Distributions](https://mlsanity.com/paper/2411.00775/) (2024)
- [Mixtures of Gaussians are Privately Learnable with a Polynomial Number of Samples](https://mlsanity.com/paper/2309.03847/) (2023)
- [SoS Certificates for Sparse Singular Values and Their Applications: Robust Statistics, Subspace Distortion, and More](https://mlsanity.com/paper/2412.21203/) (2024)
- [On Differentially Private U Statistics](https://mlsanity.com/paper/2407.04945/) (2024)
- [The Broader Landscape of Robustness in Algorithmic Statistics](https://mlsanity.com/paper/2412.02670/) (2024)
- [Tukey Depth Mechanisms for Practical Private Mean Estimation](https://mlsanity.com/paper/2502.18698/) (2025)
- [Sample-Optimal Private Regression in Polynomial Time](https://mlsanity.com/paper/2503.24321/) (2025)
- [Improved subsample-and-aggregate via the private modified winsorized mean](https://mlsanity.com/paper/2501.14095/) (2025)
- [Optimal Differentially Private Sampling of Unbounded Gaussians](https://mlsanity.com/paper/2503.01766/) (2025)
- [Private Geometric Median in Nearly-Linear Time](https://mlsanity.com/paper/2505.20189/) (2025)
- [Privately Estimating Black-Box Statistics](https://mlsanity.com/paper/2510.00322/) (2025)
- [Adaptive Power Iteration Method for Differentially Private PCA](https://mlsanity.com/paper/2602.11454/) (2026)
- [Statistical Barriers to Affine-equivariant Estimation](https://mlsanity.com/paper/2310.10758/) (2023)
- [Near-Optimal differentially private low-rank trace regression with guaranteed private initialization](https://mlsanity.com/paper/2403.15999/) (2024)
- [A Private Approximation of the 2nd-Moment Matrix of Any Subsamplable Input](https://mlsanity.com/paper/2505.14251/) (2025)
- [Differentially Private Two-Stage Gradient Descent for Instrumental Variable Regression](https://mlsanity.com/paper/2509.22794/) (2025)
- [Computationally tractable robust differentially private mean estimation](https://mlsanity.com/paper/2606.12654/) (2026)
- [Improved Private Sparse Covariance Estimation with Multiscale Threshold Tests](https://mlsanity.com/paper/2609.22783/) (2026)

## Abstract (arXiv)

We present a fast, differentially private algorithm for high-dimensional covariance-aware mean estimation with nearly optimal sample complexity. Only exponential-time estimators were previously known to achieve this guarantee. Given n samples from a (sub-)Gaussian distribution with unknown mean μ and covariance Sigma, our (varepsilon,δ)-differentially private estimator produces tildeμ such that |μ - tildeμ|Sigma ≤ α as long as n gtrsim tfrac d α2 + tfracd √log 1/δα varepsilon+fracdlog 1/δvarepsilon. The Mahalanobis error metric |μ - hatμ|Sigma measures the distance between hat μ and μ relative to Sigma; it characterizes the error of the sample mean. Our algorithm runs in time tildeO(ndω - 1 + nd/varepsilon), where ω < 2.38 is the matrix multiplication exponent. We adapt an exponential-time approach of Brown, Gaboardi, Smith, Ullman, and Zakynthinou (2021), giving efficient variants of stable mean and covariance estimation subroutines that also improve the sample complexity to the nearly optimal bound above. Our stable covariance estimator can be turned to private covariance estimation for unrestricted subgaussian distributions. With ngtrsim d3/2 samples, our estimate is accurate in spectral norm. This is the first such algorithm using n= o(d2) samples, answering an open question posed by Alabi et al. (2022). With ngtrsim d2 samples, our estimate is accurate in Frobenius norm. This leads to a fast, nearly optimal algorithm for private learning of unrestricted Gaussian distributions in TV distance. Duchi, Haque, and Kuditipudi (2023) obtained similar results independently and concurrently.