Shape And Structure Preserving Differential Privacy
Shape And Structure Preserving Differential Privacy
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DOI：
10.48550/arxiv.2209.12667
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发表时间：
2022-09
期刊：
影响因子：
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通讯作者：
Carlos Soto;K. Bharath;M. Reimherr;Aleksandra B. Slavkovic
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文献类型：
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作者：
Carlos Soto;K. Bharath;M. Reimherr;Aleksandra B. Slavkovic
通常，数据结构（如图像和2D对象的形状）被表示为流形上的点。从这些数据中产生净化的差异私人估计的机制的效用与它与空间的底层结构和几何形状的兼容性密切相关。特别是，如最近所示，效用的拉普拉斯机制上的正弯曲的流形，如肯德尔的2D形状空间，显着的曲率的影响。集中在消毒的Fr\'echet平均的一个流形上的点的样本的问题，我们利用的特征的平均值的最小化的目标函数组成的平方距离的总和，并开发一个K-范数梯度机制的黎曼流形上，有利于值，产生梯度接近零的目标函数。对于积极弯曲的流形的情况下，我们描述了如何使用平方距离函数的梯度提供更好的控制灵敏度比拉普拉斯机制，并证明了这一数值上的胼胝体的形状的数据集。进一步说明机制的效用上的一个球体和对称正定矩阵的流形。
It is common for data structures such as images and shapes of 2D objects to be represented as points on a manifold. The utility of a mechanism to produce sanitized differentially private estimates from such data is intimately linked to how compatible it is with the underlying structure and geometry of the space. In particular, as recently shown, utility of the Laplace mechanism on a positively curved manifold, such as Kendall's 2D shape space, is significantly influences by the curvature. Focusing on the problem of sanitizing the Fr\'echet mean of a sample of points on a manifold, we exploit the characterisation of the mean as the minimizer of an objective function comprised of the sum of squared distances and develop a K-norm gradient mechanism on Riemannian manifolds that favors values that produce gradients close to the the zero of the objective function. For the case of positively curved manifolds, we describe how using the gradient of the squared distance function offers better control over sensitivity than the Laplace mechanism, and demonstrate this numerically on a dataset of shapes of corpus callosa. Further illustrations of the mechanism's utility on a sphere and the manifold of symmetric positive definite matrices are also presented.