Private Approximations of a Convex Hull in Low Dimensions

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 $(1+\alpha)\mu(D_P(\kappa-\Delta))$. Our work is aimed at producing a \emph{$(\alpha,\Delta)$-kernel of $D_P(\kappa)$}, namely a set $\calS$ such that (after a shift) it holds that $(1-\alpha)D_P(\kappa)\subset \CH(\calS) \subset (1+\alpha)D_P(\kappa-\Delta)$. 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.