Shape and Structure Preserving Differential Privacy

Soto, Carlos; Bharath, Karthik; Reimherr, Matthew; Slavkovic, Aleksandra

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 influenced by the curvature. Focusing on the problem of sanitizing the Fréchet 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.

Citation

Soto, C., Bharath, K., Reimherr, M., & Slavkovic, A. (2022, November). Shape and Structure Preserving Differential Privacy. Poster presented at Thirty-sixth Conference on Neural Information Processing Systems (NeurIPS 2022), New Orleans, USA

• Presentation Conference Type: Poster
• Conference Name: Thirty-sixth Conference on Neural Information Processing Systems (NeurIPS 2022)
• Start Date: Nov 28, 2022
• End Date: Dec 9, 2022
• Deposit Date: Oct 26, 2022
• Public URL: https://nottingham-repository.worktribe.com/output/12895352
• Related Public URLs: https://nips.cc/virtual/2022/papers.html?filter=sessions&search=Poster+Session+2
• Additional Information: Tu, Nov 29, 22:00 -- Poster Session 2