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

Private Covariance Approximation and Eigenvalue-Gap Bounds for Complex Gaussian Perturbations