# Optimal Differentially Private PCA and Estimation for Spiked Covariance Matrices

arXiv: https://arxiv.org/abs/2401.03820 · HTML: https://mlsanity.com/paper/2401.03820/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 64 references that are in the map.
- First posted on arXiv 8 Jan 2024.

## Builds on (most cited first, 17 of 17)

- [RAPPOR: Randomized Aggregatable Privacy-Preserving Ordinal Response](https://mlsanity.com/paper/1407.6981/) (2014)
- [Collecting Telemetry Data Privately](https://mlsanity.com/paper/1712.01524/) (2017)
- [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)
- [Differentially Private Assouad, Fano, and Le Cam](https://mlsanity.com/paper/2004.06830/) (2020)
- [Tensor SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017)
- [Covariance-Aware Private Mean Estimation Without Private Covariance Estimation](https://mlsanity.com/paper/2106.13329/) (2021)
- [Heteroskedastic PCA: Algorithm, Optimality, and Applications](https://mlsanity.com/paper/1810.08316/) (2018)
- [DP-PCA: Statistically Optimal and Differentially Private PCA](https://mlsanity.com/paper/2205.13709/) (2022)
- [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)
- [Estimation of the number of spiked eigenvalues in a covariance matrix by bulk eigenvalue matching analysis](https://mlsanity.com/paper/2006.00436/) (2020)
- [Differentially Private Covariance Revisited](https://mlsanity.com/paper/2205.14324/) (2022)
- [Private Alternating Least Squares: Practical Private Matrix Completion with Tighter Rates](https://mlsanity.com/paper/2107.09802/) (2021)
- [Private Query Release via the Johnson-Lindenstrauss Transform](https://mlsanity.com/paper/2208.07410/) (2022)
- [Re-Analyze Gauss: Bounds for Private Matrix Approximation via Dyson Brownian Motion](https://mlsanity.com/paper/2211.06418/) (2022)
- [Optimal Estimation of Low Rank Density Matrices](https://mlsanity.com/paper/1507.05131/) (2015)

## Built on by (most cited first, 11 of 11)

- [Federated Nonparametric Hypothesis Testing with Differential Privacy Constraints: Optimal Rates and Adaptive Tests](https://mlsanity.com/paper/2406.06749/) (2024)
- [Federated PCA and Estimation for Spiked Covariance Matrices: Optimal Rates and Efficient Algorithm](https://mlsanity.com/paper/2411.15660/) (2024)
- [An Iterative Algorithm for Differentially Private k-PCA with Adaptive Noise](https://mlsanity.com/paper/2508.10879/) (2025)
- [Minimax and Adaptive Covariance Matrix Estimation under Differential Privacy](https://mlsanity.com/paper/2603.19703/) (2026)
- [On the Curse of Dimensionality in Private Sparse Covariance Estimation and PCA](https://mlsanity.com/paper/2606.21951/) (2026)
- [Gap-Free Streaming PCA Beyond Rank-One Updates: Near-Optimal Rates and Applications to Differential Privacy](https://mlsanity.com/paper/2609.26508/) (2026)
- [Near-Optimal differentially private low-rank trace regression with guaranteed private initialization](https://mlsanity.com/paper/2403.15999/) (2024)
- [Distributed quasi-Newton robust estimation under differential privacy](https://mlsanity.com/paper/2408.12353/) (2024)
- [Subspace Recovery in Winsorized PCA: Insights into Accuracy and Robustness](https://mlsanity.com/paper/2502.16391/) (2025)
- [Differentially Private Manifold Denoising](https://mlsanity.com/paper/2604.00942/) (2026)
- [Enhancing Differentially Private Mechanisms via Empirical Bayes](https://mlsanity.com/paper/2606.21107/) (2026)

## Abstract (arXiv)

Estimating a covariance matrix and its associated principal components is a fundamental problem in contemporary statistics. While optimal estimation procedures have been developed with well-understood properties, the increasing demand for privacy preservation introduces new complexities to this classical problem. In this paper, we study optimal differentially private Principal Component Analysis (PCA) and covariance estimation within the spiked covariance model. We precisely characterize the sensitivity of eigenvalues and eigenvectors under this model and establish the minimax rates of convergence for estimating both the principal components and covariance matrix. These rates hold up to logarithmic factors and encompass general Schatten norms, including spectral norm, Frobenius norm, and nuclear norm as special cases. We propose computationally efficient differentially private estimators and prove their minimax optimality for sub-Gaussian distributions, up to logarithmic factors. Additionally, matching minimax lower bounds are established. Notably, compared to the existing literature, our results accommodate a diverging rank, a broader range of signal strengths, and remain valid even when the sample size is much smaller than the dimension, provided the signal strength is sufficiently strong. Both simulation studies and real data experiments demonstrate the merits of our method.