Differentially Private Geodesic and Linear Regression

Abstract: 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 recent findings of differential privacy for the Fr\'echet mean. We demonstrate the efficacy of our methodology on the sphere, $\mbS^2\subset\mbR^3$ and, since it is general to Riemannian manifolds, the manifold of Euclidean space which simplifies geodesic regression to a case of linear regression. Our methodology is general to any Riemannian manifold and thus it is suitable for data in domains such as medical imaging and computer vision.

Overview of "Differentially Private Geodesic and Linear Regression"

The paper "Differentially Private Geodesic and Linear Regression" by Aditya Kulkarni and Carlos Soto addresses a significant challenge in modern statistical methodologies: extending regression techniques to data residing on Riemannian manifolds while ensuring differential privacy (DP). Traditional linear regression assumes data lying in Euclidean spaces; however, contemporary applications frequently involve data in non-Euclidean domains like spherical or manifold-structured spaces, necessitating geodesic regression. The authors propose a novel approach to achieve differential privacy in geodesic regression by leveraging the K-Norm Gradient (KNG) mechanism. This research advances the understanding and practical implementation of privacy-preserving statistical methods on manifolds, contributing both theoretical insights and practical methodologies.

Methodological Innovations

The study explores the differential privacy of geodesic regression parameters which are not trivial due to their manifold-based nature. The authors utilize the KNG mechanism, extending its applicability to Riemannian manifolds. This involves deriving theoretical bounds for the sensitivity of geodesic regression parameters, demonstrating that they are intricately linked to Jacobi fields and manifold curvature.

The paper presents:

1. Geodesic Regression Extension: The conventional linear regression paradigm is extended to accommodate geodesic regression, where response variables reside on manifolds. The core challenge addressed is the differential privacy of the geodesic regression parameters: the footpoint and the shooting vector.

2. Differential Privacy Mechanism for Manifolds: The KNG mechanism is applied to Riemannian manifolds for the first time, ensuring the parameters of geodesic regression satisfy pure DP. This requires a deep integration of differential geometry concepts, like Jacobi fields, into the analysis.

3. Empirical Validation: The methodology is empirically validated on a unit sphere ($S^2 \subset \mathbb{R}^3$), illustrating its utility and performance. Additionally, it evaluates DP linear regression by exemplifying the special case where geodesic regression reduces to linear regression in Euclidean settings, testing against benchmark tools like IBM Diffprivlib and META Opacus.

Results and Implications

The authors showcase robust numerical results and verify their approach through simulations. Highlights include:

• Theoretical Sensitivity Bounds: The sensitivity bound derived is tighter, supporting improved utility over traditional mechanisms like the Laplace mechanism, especially in contexts with curvature constraints.
• Comparative Analysis: Against existing DP tools, the proposed method achieves better or equivalent performance in preserving data utility while ensuring differential privacy.
• Scalability and Generality: The approach is generalizable to any Riemannian manifold, suggesting applications in domains like medical imaging and computer vision, where data are naturally manifold-valued.

Future Directions

The paper opens several avenues for future research:

• Algorithmic Development for Efficient Sampling: The exploration of more efficient algorithms for sampling under the KNG mechanism in manifold settings is encouraged, potentially enhancing computational feasibility and scalability.
• Application to Diverse Manifolds: Extending the methodology to other manifolds such as the complex projective spaces associated with Kendall's shape space, which has implications for fields like shape analysis and computer vision.
• Integration with Advanced Machine Learning Models: Combining differential privacy with machine learning models operating on manifold-valued data could cultivate richer, privacy-preserving analytic techniques in AI.

In summary, the paper makes a substantial contribution to the field of differentially private statistical methods by extending regression analysis to non-linear manifolds and ensuring privacy via a novel mechanism. This sets a foundation for further advancements in privacy-aware statistical analyses on complex data structures.