Shape And Structure Preserving Differential Privacy
- URL: http://arxiv.org/abs/2209.12667v1
- Date: Wed, 21 Sep 2022 18:14:38 GMT
- Title: Shape And Structure Preserving Differential Privacy
- Authors: Carlos Soto and Karthik Bharath and Matthew Reimherr and Aleksandra
Slavkovic
- Abstract summary: We show how the gradient of the squared distance function offers better control over sensitivity than the Laplace mechanism.
We also show how using the gradient of the squared distance function offers better control over sensitivity than the Laplace mechanism.
- Score: 70.08490462870144
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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