We consider the problem of approximating a $d \times d$ covariance matrix $M$ with a rank-$k$ matrix under $(\varepsilon,\delta)$-differential privacy. We present and analyze a complex variant of the Gaussian mechanism and show that the Frobenius norm of the difference between the matrix output by this mechanism and the best rank-$k$ approximation to $M$ is bounded by roughly $\tilde{O}(\sqrt{kd})$, whenever there is an appropriately large gap between the $k$'th and the $k+1$'th eigenvalues of $M$. This improves on previous work that requires that the gap between every pair of top-$k$ eigenvalues of $M$ is at least $\sqrt{d}$ for a similar bound. Our analysis leverages the fact that the eigenvalues of complex matrix Brownian motion repel more than in the real case, and uses Dyson's stochastic differential equations governing the evolution of its eigenvalues to show that the eigenvalues of the matrix $M$ perturbed by complex Gaussian noise have large gaps with high probability. Our results contribute to the analysis of low-rank approximations under average-case perturbations and to an understanding of eigenvalue gaps for random matrices, which may be of independent interest.

翻译：我们考虑在$(\varepsilon,\delta)$-差分隐私约束下，使用秩-$k$矩阵近似$d \times d$协方差矩阵$M$的问题。我们提出并分析了高斯机制的复变体，并证明：当$M$的第$k$个与第$k+1$个特征值之间存在适当大的间隙时，该机制输出的矩阵与$M$的最优秩-$k$近似之间的Frobenius范数大致以$\tilde{O}(\sqrt{kd})$为界。这改进了先前工作中要求$M$的前$k$个特征值中每一对之间的间隙至少为$\sqrt{d}$才能获得类似界的结果。我们的分析利用了复矩阵布朗运动的特征值比实情况下的特征值具有更强的排斥性这一事实，并借助Dyson随机微分方程（该方程控制其特征值的演化）证明：经复高斯噪声扰动后的矩阵$M$的特征值以高概率具有大间隙。我们的研究结果有助于分析平均情况扰动下的低秩近似问题，并增进对随机矩阵特征值间隙的理解，这些成果可能具有独立的研究价值。