In this paper, we identify that the classic Gaussian mechanism and its variants for differential privacy all suffer from \textbf{the curse of full-rank covariance matrices}, and hence the expected accuracy losses of these mechanisms applied to high dimensional query results, e.g., in $\mathbb{R}^M$, all increase linearly with $M$. To lift this curse, we design a Rank-1 Singular Multivariate Gaussian Mechanism (R1SMG). It achieves $(\epsilon,\delta)$-DP on query results in $\mathbb{R}^M$ by perturbing the results with noise following a singular multivariate Gaussian distribution, whose covariance matrix is a \textbf{randomly} generated rank-1 positive semi-definite matrix. In contrast, the classic Gaussian mechanism and its variants all consider \textbf{deterministic} full-rank covariance matrices. Our idea is motivated by a clue from Dwork et al.'s work on Gaussian mechanism that has been ignored in the literature: when projecting multivariate Gaussian noise with a full-rank covariance matrix onto a set of orthonormal basis in $\mathbb{R}^M$, only the coefficient of a single basis can contribute to the privacy guarantee. This paper makes the following technical contributions. (i) R1SMG achieves $(\epsilon,\delta)$-DP guarantee on query results in $\mathbb{R}^M$, while the magnitude of the additive noise decreases with $M$. Therefore, \textbf{less is more}, i.e., less amount of noise is able to sanitize higher dimensional query results. When $M\rightarrow \infty$, the expected accuracy loss converges to ${2(\Delta_2f)^2}/{\epsilon}$, where $\Delta_2f$ is the $l_2$ sensitivity of the query function $f$. (ii) Compared with other mechanisms, R1SMG is less likely to generate noise with large magnitude that overwhelms the query results, because the kurtosis and skewness of the nondeterministic accuracy loss introduced by R1SMG is larger than that introduced by other mechanisms.

翻译：本文指出现有经典高斯机制及其变体均受困于**满秩协方差矩阵的维度灾难**，导致这些机制应用于高维查询结果（如$\mathbb{R}^M$空间）时的期望精度损失均随$M$线性增长。为突破这一局限，我们设计了秩-1奇异多元高斯机制（R1SMG）。该机制通过采用协方差矩阵为**随机**生成的秩-1半正定矩阵的奇异多元高斯分布噪声扰动查询结果，在$\mathbb{R}^M$中实现$(\epsilon,\delta)$-差分隐私。而经典高斯机制及其变体均采用**确定性**满秩协方差矩阵。这一设计灵感源于Dwork等人在高斯机制中一个被文献忽略的线索：将满秩协方差矩阵的多元高斯噪声投影到$\mathbb{R}^M$的一组标准正交基时，仅单个基的系数可贡献隐私保证。本文的技术贡献包括：（i）R1SMG在$\mathbb{R}^M$中实现$(\epsilon,\delta)$-DP保证，且加性噪声幅度随$M$递减。因此**少即是多**，即更少的噪声量即可净化更高维的查询结果。当$M\rightarrow\infty$时，期望精度损失收敛于${2(\Delta_2f)^2}/{\epsilon}$，其中$\Delta_2f$为查询函数$f$的$l_2$敏感度。（ii）与其他机制相比，R1SMG更不易产生淹没查询结果的大幅度噪声，因其引入的非确定性精度损失具有更高的峰度与偏度。