Improved Private Sparse Covariance Estimation with Multiscale Threshold Tests

arXiv:2609.22783v1 Announce Type: new Abstract: We study differentially private covariance estimation in operator norm for mean-zero sub-Gaussian distributions with unknown covariance support and at most $k$ nonzero entries per row. We develop a multiscale random-threshold algorithm with sample complexity $\ot(k^2/\alpha^2+k\sqrt d/(\alpha\varepsilon))$ for $(\varepsilon,\delta)$-differential privacy and error at most $\alpha\sigma^2$, where $d$ is the dimension and $\sigma$ is a known sub-Gaussian scale. The bound improves the privacy-dependent term of the existing $\ot(k^2/\alpha^2+k^{3/2}\sqrt d/(\alpha\varepsilon))$ \citep{kumar2026curse} upper bound by a factor of $\sqrt k$, and matches the lower bound of $\widetilde{\Omega}(k^2/\alpha^2 + k\sqrt{d}/(\alpha\varepsilon))$ in its applicable parameter regime. Our key technical ingredient is a direct operator-norm bound on the centered fluctuations of an ideal reconstruction, exploiting conditional independence rather than accumulating entrywise errors across each row. A multiscale allocation of threshold tests balances reconstruction variance against query sensitivity. Together, these ingredients sharpen the trade-off between approximation error and privacy protection, removing the additional $\sqrt{k}$ factor from the privacy-dependent sample complexity.