Differential Privacy Over Riemannian Manifolds
Differential Privacy Over Riemannian Manifolds
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发表时间：
2021-11
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通讯作者：
M. Reimherr;K. Bharath;Carlos Soto
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作者：
M. Reimherr;K. Bharath;Carlos Soto
在这项工作中，我们考虑的问题，释放一个差分私人统计摘要，驻留在黎曼流形。我们提出了一个扩展的拉普拉斯或K-范数机制，利用内在的距离和体积的流形。我们还考虑了详细的具体情况下，摘要是Fr\'echet平均驻留在流形上的数据。我们证明了我们的机制是速率最优的，并且仅取决于流形的维度，而不取决于任何周围空间的维度，同时还展示了忽略流形结构如何降低消毒摘要的效用。我们在统计学中特别感兴趣的两个例子中说明了我们的框架：用于协方差矩阵的对称正定矩阵空间，以及可用作离散分布建模空间的球体。
In this work we consider the problem of releasing a differentially private statistical summary that resides on a Riemannian manifold. We present an extension of the Laplace or K-norm mechanism that utilizes intrinsic distances and volumes on the manifold. We also consider in detail the specific case where the summary is the Fr\'echet mean of data residing on a manifold. We demonstrate that our mechanism is rate optimal and depends only on the dimension of the manifold, not on the dimension of any ambient space, while also showing how ignoring the manifold structure can decrease the utility of the sanitized summary. We illustrate our framework in two examples of particular interest in statistics: the space of symmetric positive definite matrices, which is used for covariance matrices, and the sphere, which can be used as a space for modeling discrete distributions.