Differentially Private Covariance Revisited
Authors: Wei Dong, Yuting Liang, Ke Yi

In this paper, we present two new algorithms for covariance estimation under concentrated differential privacy (zCDP).  The first algorithm achieves a Frobenius error of $\tilde{O}(d^{1/4}\sqrt{\mathrm{tr}}/\sqrt{n} + \sqrt{d}/n)$, where $\mathrm{tr}$ is the trace of the covariance matrix.  By taking $\mathrm{tr}=1$, this also implies a worst-case error bound of $\tilde{O}(d^{1/4}/\sqrt{n})$, which improves the standard Gaussian mechanism's $\tilde{O}(d/n)$ for the regime $d>\widetilde{\Omega}(n^{2/3})$.  Our second algorithm offers a tail-sensitive bound that could be much better on skewed data.  The corresponding algorithms are also simple and efficient. Experimental results show that they offer significant improvements over prior work.