# Minimax and Adaptive Covariance Matrix Estimation under Differential Privacy

arXiv: https://arxiv.org/abs/2603.19703 · HTML: https://mlsanity.com/paper/2603.19703/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 45 references that are in the map.
- First posted on arXiv 20 Mar 2026.

## Builds on (most cited first, 20 of 20)

- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [Concentrated Differential Privacy](https://mlsanity.com/paper/1603.01887/) (2016)
- [Differentially Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds](https://mlsanity.com/paper/1405.7085/) (2014)
- [Minimax Optimal Procedures for Locally Private Estimation](https://mlsanity.com/paper/1604.02390/) (2016)
- [Privately Learning High-Dimensional Distributions](https://mlsanity.com/paper/1805.00216/) (2018)
- [CoinPress: Practical Private Mean and Covariance Estimation](https://mlsanity.com/paper/2006.06618/) (2020)
- [Differential privacy and robust statistics in high dimensions](https://mlsanity.com/paper/2111.06578/) (2021)
- [Differentially Private Assouad, Fano, and Le Cam](https://mlsanity.com/paper/2004.06830/) (2020)
- [Private and polynomial time algorithms for learning Gaussians and beyond](https://mlsanity.com/paper/2111.11320/) (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)
- [Local Privacy, Data Processing Inequalities, and Statistical Minimax Rates](https://mlsanity.com/paper/1302.3203/) (2013)
- [Score Attack: A Lower Bound Technique for Optimal Differentially Private Learning](https://mlsanity.com/paper/2303.07152/) (2023)
- [Better and Simpler Lower Bounds for Differentially Private Statistical Estimation](https://mlsanity.com/paper/2310.06289/) (2023)
- [Differentially Private Covariance Revisited](https://mlsanity.com/paper/2205.14324/) (2022)
- [Optimal Differentially Private PCA and Estimation for Spiked Covariance Matrices](https://mlsanity.com/paper/2401.03820/) (2024)
- [Optimal Federated Learning for Nonparametric Regression with Heterogeneous Distributed Differential Privacy Constraints](https://mlsanity.com/paper/2406.06755/) (2024)
- [Lower Bounds for Private Estimation of Gaussian Covariance Matrices under All Reasonable Parameter Regimes](https://mlsanity.com/paper/2404.17714/) (2024)
- [Smooth Lower Bounds for Differentially Private Algorithms via Padding-and-Permuting Fingerprinting Codes](https://mlsanity.com/paper/2307.07604/) (2023)
- [Optimal estimation in private distributed functional data analysis](https://mlsanity.com/paper/2412.06582/) (2024)

## Built on by (most cited first, 1 of 1)

- [A Van Trees Lower Bound for Fully Interactive Differentially Private Federated Learning](https://mlsanity.com/paper/2605.19813/) (2026)

## Abstract (arXiv)

Estimating covariance matrices is fundamental to a wide range of statistical applications. This paper studies minimax and adaptive estimation of high-dimensional covariance matrices under ρ-zero-concentrated differential privacy (ρ-zCDP) over three nested classes: the pointwise-decay class Hα, the row-tail class Gα, and the separated-block class Fα. We consider both squared operator norm loss and normalized squared Frobenius norm loss. For Hα and Gα, we develop center--outer dyadic estimators tailored to the refined geometry of the two classes, while for Fα, we develop a blockwise tridiagonal estimator. The resulting minimax-optimal rates reveal a nontrivial interplay among the smoothness α, the loss, the geometry of the covariance class, and the privacy constraint. In contrast to the non-private setting, privacy distinguishes covariance classes that share the same leading non-private rate and induces a polynomial dependence on the ambient dimension. We further develop procedures that adapt to the unknown decay parameter over all three covariance classes under both losses, at the cost of at most polylogarithmic factors. To establish minimax lower bounds, we develop a novel differentially private van Trees inequality that connects Fisher information with the ρ-zCDP constraint and may be useful for other private estimation problems. We also construct carefully designed prior distributions to obtain matching minimax lower bounds.