Skip to main content

Research Repository

Advanced Search

Shape and Structure Preserving Differential Privacy

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


Carlos Soto

Matthew Reimherr

Aleksandra Slavkovic


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.


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)
Conference Location New Orleans, USA
Start Date Nov 28, 2022
End Date Dec 9, 2022
Deposit Date Oct 26, 2022
Public URL
Related Public URLs
Additional Information Tu, Nov 29, 22:00 -- Poster Session 2