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 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 (\alpha,\Delta)-kernel of D_P(\kappa), namely a set %DIFDELCMD < \calS %%%  such that (after a shift) it holds that (1-\alpha)D_P(\kappa)\subset %DIFDELCMD < \CH%%% (%DIFDELCMD < \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], 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 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.
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 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 (\alpha,\Delta)-kernel of D_P(\kappa), namely a set %DIFDELCMD < \calS %%% \mathcal{S such that (after a shift) it holds that (1-\alpha)D_P(\kappa)\subset %DIFDELCMD < \CH%%% %DIFDELCMD < \calS%%% \mathsf{CH ) \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], 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 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.
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D
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A
\mathsf{CH