Differentially 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 Tukey-depth. Instead of (non-privately) approximating the convex-hull of the given set of points P, our algorithms approximate the geometric features of DP pκq - the κ-Tukey region induced by P (all points of Tukey-depth κ or greater). Moreover, our approximations are all bi-criteria: for any geometric feature µ our pα, ∆q-approximation is a value “sandwiched” between p1 αqµpDP pκqq and p1 αqµpDP pκ ∆qq. Our work is aimed at producing a pα, ∆q-kernel of DP pκq, namely a set S such that (after a shift) it holds that p1 αqDP pκq Ă CHpSq Ă p1 αqDP pκ ∆q. We show that an analogous notion of a bi-critera approximation of a directional kernel, as originally proposed by [1], 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 pα, ∆q-kernels for a “fat” Tukey-region. Then, based on a private approximation of the min-bounding box, we find a transformation that does turn DP pκq into a “fat” region but only if its volume is proportional to the volume of DP pκ ∆q. Lastly, we give a novel private algorithm that finds a depth parameter κ for which the volume of DP pκq is comparable to the volume of DP pκ ∆q. We hope our work leads to the further study of the intersection of differential privacy and computational geometry.