发表机构
HKUST（香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对次高斯分布私有稀疏协方差估计，提出多尺度随机阈值算法，将隐私相关样本复杂度从 $k^{3/2}$ 降至 $k$，匹配下界并消除 $\sqrt{k}$ 因子。
AI 中文摘要
我们研究了在算子范数下，对于均值为零、具有未知协方差支撑且每行至多有 $k$ 个非零项的次高斯分布，其差分私有协方差估计问题。我们开发了一种多尺度随机阈值算法，其样本复杂度为 $\ot(k^2/\alpha^2+k\sqrt d/(\alpha\varepsilon))$，满足 $(\varepsilon,\delta)$-差分隐私，且误差至多为 $\alpha\sigma^2$，其中 $d$ 是维度，$\sigma$ 是已知的次高斯尺度。该界将现有 $\ot(k^2/\alpha^2+k^{3/2}\sqrt d/(\alpha\varepsilon))$ \citep{kumar2026curse} 上界中依赖隐私的项改进了一个 $\sqrt k$ 因子，并在其适用参数范围内匹配了下界 $\widetilde{\Omega}(k^2/\alpha^2 + k\sqrt{d}/(\alpha\varepsilon))$。我们的关键技术要素是对理想重建的中心波动直接给出算子范数界，利用条件独立性而非逐行累积逐项误差。多尺度阈值检验的分配在重建方差与查询灵敏度之间取得平衡。这些要素共同优化了近似误差与隐私保护之间的权衡，从依赖隐私的样本复杂度中消除了额外的 $\sqrt{k}$ 因子。
英文摘要
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 $\ot(k^2/α^2+k\sqrt d/(α\varepsilon))$ for $(\varepsilon,δ)$-differential privacy and error at most $ασ^2$, where $d$ is the dimension and $σ$ is a known sub-Gaussian scale. The bound improves the privacy-dependent term of the existing $\ot(k^2/α^2+k^{3/2}\sqrt d/(α\varepsilon))$ \citep{kumar2026curse} upper bound by a factor of $\sqrt k$, and matches the lower bound of $\widetildeΩ(k^2/α^2 + k\sqrt{d}/(α\varepsilon))$ 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.
