Private Approximations of a Convex Hull in Low Dimensions

Abstract

We give the first differentially private algorithms that estimate a variety of geometric features of points in the Euclidean space, such as diameter, width, volume of convex hull, min-bounding box, min-enclosing ball etc. Our work relies heavily on the notion of \\emph{Tukey-depth}. Instead of (non-privately) approximating the convex-hull of the given set of points $P$, our algorithms approximate the geometric features of the $\\kappa$-Tukey region induced by $P$ (all points of Tukey-depth $\\kappa$ or greater). Moreover, our approximations are all bi-criteria: for any geometric feature $\\mu$ our $(\\alpha,\\Delta)$-approximation is a value sandwiched between $(1-\\alpha)\\mu(D_P(\\kappa))$ and $(1+\\alpha)\\mu(D_P(\\kappa-\\Delta))$. \nOur work is aimed at producing a \\emph{$(\\alpha,\\Delta)$-kernel of $D_P(\\kappa)$}, namely a set $\\mathcal{S}$ such that (after a shift) it holds that $(1-\\alpha)D_P(\\kappa)\\subset \\mathsf{CH}(\\mathcal{S}) \\subset (1+\\alpha)D_P(\\kappa-\\Delta)$. We show that an analogous notion of a bi-critera approximation of a directional kernel, as originally proposed by Agarwal et al~[2004], \\emph{fails} to give a kernel, and so we result to subtler notions of approximations of projections that do yield a kernel. First, we give differentially private algorithms that find $(\\alpha,\\Delta)$-kernels for a fat Tukey-region. Then, based on a private approximation of the min-bounding box, we find a transformation that does turn $D_P(\\kappa)$ into a fat region \\emph{but only if} its volume is proportional to the volume of $D_P(\\kappa-\\Delta)$. Lastly, we give a novel private algorithm that finds a depth parameter $\\kappa$ for which the volume of $D_P(\\kappa)$ is comparable to $D_P(\\kappa-\\Delta)$. We hope this work leads to the further study of the intersection of differential privacy and computational geometry.