Differential privacy via output perturbation has been a de facto standard for releasing query or computation results on sensitive data. However, we identify that all existing Gaussian mechanisms suffer from the curse of full-rank covariance matrices. To lift this curse, we design a Rank-1 Singular Multivariate Gaussian (R1SMG) mechanism. It achieves DP on high dimension query results by perturbing the results with noise following a singular multivariate Gaussian distribution, whose covariance matrix is a randomly generated rank-1 positive semi-definite matrix. In contrast, the classic Gaussian mechanism and its variants all consider deterministic full-rank covariance matrices. Our idea is motivated by a clue from Dwork et al.'s seminal work on the classic 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, only the coefficient of a single basis can contribute to the privacy guarantee. This paper makes the following technical contributions. The R1SMG mechanisms achieves DP guarantee on high dimension query results, while its expected accuracy loss is lower bounded by a term that is on a lower order of magnitude by at least the dimension of query results compared existing Gaussian mechanisms. Compared with other mechanisms, the R1SMG mechanism is more stable and 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 this mechanism is larger than that introduced by other mechanisms.

翻译：通过输出扰动实现差分隐私已成为在敏感数据上发布查询或计算结果的事实标准。然而，我们发现所有现有的高斯机制都受到满秩协方差矩阵的维数灾难困扰。为解除这一困境，我们设计了一种秩一奇异多元高斯（R1SMG）机制。该机制通过使用服从奇异多元高斯分布的噪声扰动结果来实现高维查询结果的差分隐私，其协方差矩阵为随机生成的秩一正半定矩阵。相比之下，经典高斯机制及其变体均采用确定性满秩协方差矩阵。我们的想法源于Dwork等人在经典高斯机制开创性工作中被文献忽略的线索：当将具有满秩协方差矩阵的多元高斯噪声投影到一组标准正交基上时，仅有单个基的系数能贡献于隐私保证。本文做出以下技术贡献：R1SMG机制实现了高维查询结果的差分隐私保证，同时其期望精度损失的下界项在数量级上比现有高斯机制至少低一个维度量级。与其他机制相比，R1SMG机制更稳定且不易产生淹没查询结果的大幅值噪声，这是因为该机制引入的非确定性精度损失的峰度和偏度均大于其他机制。