We study the problem of estimating the spectral density of a centered stationary Gaussian time series under local differential privacy constraints. Specifically, we propose new interactive privacy mechanisms for three tasks: recovering a single covariance coefficient, recovering the spectral density at a fixed frequency, and global recovery. Our approach achieves faster rates through a two-stage process: we first apply the Laplace mechanism to the truncated value, and then use the resulting privatized sample to learn about the dependence mechanism in the time series. For spectral densities belonging to Hölder and Sobolev smoothness classes, we demonstrate that our algorithms improve upon the non-interactive mechanism of Kroll (2024) for small privacy parameter $α$, since the pointwise rates depend on $nα^2$ instead of $nα^4$. Moreover, we show that the rate $(nα^4)^{-1}$ is optimal for estimating a covariance coefficient with non-interactive mechanisms. However, the $L_2$ rate of our interactive estimator is slower than the pointwise rate. We show how to use these procedures to provide a bona fide locally differentially private estimator of the entire covariance matrix. A simulation study validates our findings.

翻译：本文研究了在局部差分隐私约束下估计中心化平稳高斯时间序列谱密度的问题。具体而言，我们针对三项任务提出了新的交互式隐私机制：恢复单个协方差系数、恢复固定频率处的谱密度以及全局恢复。我们的方法通过两阶段过程实现了更快的收敛速率：首先对截断值应用拉普拉斯机制，然后利用得到的隐私化样本来学习时间序列中的依赖机制。对于属于Hölder和Sobolev光滑性类的谱密度，我们证明了当隐私参数$α$较小时，我们的算法优于Kroll（2024）的非交互式机制，因为逐点速率依赖于$nα^2$而非$nα^4$。此外，我们证明了$(nα^4)^{-1}$这一速率对于使用非交互式机制估计协方差系数是最优的。然而，我们交互式估计量的$L_2$速率慢于逐点速率。我们展示了如何利用这些程序构建整个协方差矩阵的严格局部差分隐私估计量。模拟研究验证了我们的结论。