论文标题
差异私有的fréchet平均值在对称阳性（SPD）矩阵的流形上
Differentially Private Fréchet Mean on the Manifold of Symmetric Positive Definite (SPD) Matrices with log-Euclidean Metric
论文作者
论文摘要
差异隐私对于具有严格的隐私保证的统计和机器学习算法的现实部署至关重要。已经开发了差异隐私机制的最早的统计查询是释放样本均值。在几何统计中，样品fréchet平均值代表了最基本的统计摘要之一，因为它概括了属于非线性歧管的数据的样本均值。本着这种精神，迄今为止，已经开发出差异隐私机制的唯一几何统计查询是释放样本fréchet的平均值：\ emph {riemannian laplace机制}最近被提议私有化在完整的riemannian杂志上的fréchet均值。在许多领域中，对称正定（SPD）矩阵的流形用于对数据空间进行建模，包括在隐私要求是关键的医学成像中。我们提出了一种新颖，简单且快速的机制 - \ emph {切线高斯机构} - 以计算带有lio -euclidean riemannian指标的SPD歧管上的差异私有fréchet。我们表明，我们的新机制具有更好的实用性，并且在计算上是有效的 - 如广泛的实验所证实。
Differential privacy has become crucial in the real-world deployment of statistical and machine learning algorithms with rigorous privacy guarantees. The earliest statistical queries, for which differential privacy mechanisms have been developed, were for the release of the sample mean. In Geometric Statistics, the sample Fréchet mean represents one of the most fundamental statistical summaries, as it generalizes the sample mean for data belonging to nonlinear manifolds. In that spirit, the only geometric statistical query for which a differential privacy mechanism has been developed, so far, is for the release of the sample Fréchet mean: the \emph{Riemannian Laplace mechanism} was recently proposed to privatize the Fréchet mean on complete Riemannian manifolds. In many fields, the manifold of Symmetric Positive Definite (SPD) matrices is used to model data spaces, including in medical imaging where privacy requirements are key. We propose a novel, simple and fast mechanism - the \emph{tangent Gaussian mechanism} - to compute a differentially private Fréchet mean on the SPD manifold endowed with the log-Euclidean Riemannian metric. We show that our new mechanism has significantly better utility and is computationally efficient -- as confirmed by extensive experiments.
