Improved Private Sparse Covariance Estimation with Multiscale Threshold Tests

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(k2/α2+k√ d/(αvarepsilon)) for (varepsilon,δ)-differential privacy and error at most ασ2, where d is the dimension and σ is a known sub-Gaussian scale. The bound improves the privacy-dependent term of the existing ot(k2/α2+k3/2√ d/(αvarepsilon)) citepkumar2026curse upper bound by a factor of √ k, and matches the lower bound of widetildeOmega(k2/α2 + k√d/(αvarepsilon)) 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 √k factor from the privacy-dependent sample complexity.