We present a simple perturbation mechanism for the release of $d$-dimensional covariance matrices $\Sigma$ under pure differential privacy. For large datasets with at least $n\geq d^2/\varepsilon$ elements, our mechanism recovers the provably optimal Frobenius norm error guarantees of \cite{nikolov2023private}, while simultaneously achieving best known error for all other $p$-Schatten norms, with $p\in [1,\infty]$. Our error is information-theoretically optimal for all $p\ge 2$, in particular, our mechanism is the first purely private covariance estimator that achieves optimal error in spectral norm. For small datasets $n< d^2/\varepsilon$, we further show that by projecting the output onto the nuclear norm ball of appropriate radius, our algorithm achieves the optimal Frobenius norm error $O(\sqrt{d\;\text{Tr}(\Sigma) /n})$, improving over the known bounds of $O(\sqrt{d/n})$ of \cite{nikolov2023private} and ${O}\big(d^{3/4}\sqrt{\text{Tr}(\Sigma)/n}\big)$ of \cite{dong2022differentially}.

翻译：我们提出了一种简单的扰动机制，用于在纯差分隐私下发布$d$维协方差矩阵$\Sigma$。对于元素数量至少为$n\geq d^2/\varepsilon$的大规模数据集，我们的机制恢复了\cite{nikolov2023private}中证明最优的Frobenius范数误差保证，同时对所有其他$p$-Schatten范数（$p\in [1,\infty]$）实现了目前已知的最佳误差。对于所有$p\ge 2$，我们的误差在信息论意义下是最优的；特别地，我们的机制是首个在谱范数下达到最优误差的纯私有协方差估计器。对于小规模数据集$n< d^2/\varepsilon$，我们进一步证明，通过将输出投影到适当半径的核范数球上，我们的算法实现了最优Frobenius范数误差$O(\sqrt{d\;\text{Tr}(\Sigma) /n})$，改进了\cite{nikolov2023private}中$O(\sqrt{d/n})$和\cite{dong2022differentially}中${O}\big(d^{3/4}\sqrt{\text{Tr}(\Sigma)/n}\big)$的已知误差界。