Carlos Soto (UMass Amherst): Differential Privacy over Riemannian Manifolds and Shape Space

Carlos Soto (UMass Amherst)

Title: Differential Privacy over Riemannian Manifolds and Shape Space

Abstract: Motivated by the problem of statistical shape analysis, this work considers the problem of producing sanitised differentially private estimates through the K-norm Gradient Mechanism (KNG) when the data or parameters live on a Riemannian manifold. In particular, Kendall's 2D shape space is a Riemannian manifold (a projective space) with positive sectional curvature. Traditionally, KNG requires an objective function and produces a sanitized estimate by favoring values which produce gradients close to zero of the objective. This work extends KNG to consider objective functions which take on manifold-valued data. Respecting the nature of the data leads to utility gains when compared to sanitization in an ambient space, as well as removing the need for post-processing. Specifically, this work proposes sanitizing the Fréchet mean for the sphere, symmetric positive definite matrices, and Kendall’s 2D shape space under a pure differentially private framework with an application to corpus callosum data.