Shape And Structure Preserving Differential Privacy

TL;DR

This paper introduces a geometry-aware differential privacy mechanism for data on manifolds, improving utility in shape analysis by leveraging Riemannian geometry and gradient-based sensitivity control.

Contribution

It develops a K-norm gradient mechanism tailored for Riemannian manifolds, enhancing privacy-utility trade-offs for shape and structure data.

Findings

The mechanism outperforms Laplace in positively curved manifolds.

Numerical results on shape data demonstrate improved privacy utility.

Applications include shapes of corpus callosa, spheres, and positive definite matrices.

Abstract

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.