We study elliptical distributions in locally convex vector spaces, and determine conditions when they can or cannot be used to satisfy differential privacy (DP). A requisite condition for a sanitized statistical summary to satisfy DP is that the corresponding privacy mechanism must induce equivalent measures for all possible input databases. We show that elliptical distributions with the same dispersion operator, $C$, are equivalent if the difference of their means lies in the Cameron-Martin space of $C$. In the case of releasing finite-dimensional projections using elliptical perturbations, we show that the privacy parameter $\ep$ can be computed in terms of a one-dimensional maximization problem. We apply this result to consider multivariate Laplace, $t$, Gaussian, and $K$-norm noise. Surprisingly, we show that the multivariate Laplace noise does not achieve $\ep$-DP in any dimension greater than one. Finally, we show that when the dimension of the space is infinite, no elliptical distribution can be used to give $\ep$-DP; only $(\epsilon,\delta)$-DP is possible.
翻译:我们研究本地锥体矢量空间的椭圆分布,确定它们能够或不能用于满足不同隐私(DP)的条件。 清洁统计摘要满足DP的一个必要条件是,相应的隐私机制必须为所有可能的输入数据库引入等量措施。 我们显示,如果其手段的差别在于卡梅伦-马丁空间$C美元,则与同一分散操作器($C美元)的椭圆分布相当。 在使用椭圆扰动释放有限维度预测的情况下,我们显示,可以用一维最大化问题来计算隐私参数$/ep$。我们应用这一结果来考虑多变拉普尔、$t$、Gaussian和$K-noum噪音。令人惊讶的是,我们显示,多变拉普尔噪音在任何比一维都大于1美元的范围内都达不到$\ep-DP美元。 最后,我们显示,当空间的维度是无限的时,不能使用螺旋分布来提供$ep-DP美元; 只有$\epslon, 可能使用$-DP。