In statistical applications it has become increasingly common to encounter data structures that live on non-linear spaces such as manifolds. Classical linear regression, one of the most fundamental methodologies of statistical learning, captures the relationship between an independent variable and a response variable which both are assumed to live in Euclidean space. Thus, geodesic regression emerged as an extension where the response variable lives on a Riemannian manifold. The parameters of geodesic regression, as with linear regression, capture the relationship of sensitive data and hence one should consider the privacy protection practices of said parameters. We consider releasing Differentially Private (DP) parameters of geodesic regression via the K-Norm Gradient (KNG) mechanism for Riemannian manifolds. We derive theoretical bounds for the sensitivity of the parameters showing they are tied to their respective Jacobi fields and hence the curvature of the space. This corroborates, and extends, recent findings of differential privacy for the Fréchet mean. We demonstrate the efficacy of our methodology on the sphere, $S_2\subset\mathbb{R}^3$, the space of symmetric positive definite matrices, and Kendall's planar shape space. Our methodology is general to any Riemannian manifold, and thus it is suitable for data in domains such as medical imaging and computer vision.

翻译：在统计应用中，处理定义在非线性空间（如流形）上的数据结构已日益普遍。经典线性回归作为统计学习最基础的方法之一，用于刻画自变量与响应变量之间的关系，但二者通常被假定存在于欧氏空间中。因此，测地回归应运而生，它将响应变量扩展至黎曼流形上。与线性回归类似，测地回归的参数反映了敏感数据间的关联，因此需要考虑对这些参数的隐私保护措施。本文研究通过黎曼流形上的K-范数梯度（KNG）机制发布测地回归的差分隐私（DP）参数。我们推导了参数敏感度的理论界，证明其与相应的雅可比场及空间的曲率密切相关。这一结论证实并扩展了最近关于弗雷歇均值的差分隐私研究结果。我们在球面$S_2\subset\mathbb{R}^3$、对称正定矩阵空间以及肯德尔平面形状空间上验证了方法的有效性。本方法适用于任意黎曼流形，因此可广泛应用于医学影像与计算机视觉等领域的数据处理。