eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-07-19
18:1
18:16
10.4230/LIPIcs.ITC.2021.18
article
Differentially Private Approximations of a Convex Hull in Low Dimensions
Gao, Yue
1
Sheffet, Or
2
https://orcid.org/0000-0002-5182-0530
Department of Computing Science, University of Alberta, Edmonton, Canada
Faculty of Engineering, Bar-Ilan University, Ramat Gan, Israel
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 D_{P}(κ) - 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 (α,Δ)-approximation is a value "sandwiched" between (1-α)μ(D_P(κ)) and (1+α)μ(D_P(κ-Δ)).
Our work is aimed at producing a (α,Δ)-kernel of D_P(κ), namely a set 𝒮 such that (after a shift) it holds that (1-α)D_P(κ) ⊂ CH(𝒮) ⊂ (1+α)D_P(κ-Δ). We show that an analogous notion of a bi-critera approximation of a directional kernel, as originally proposed by [Pankaj K. 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 (α,Δ)-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(κ) into a "fat" region but only if its volume is proportional to the volume of D_P(κ-Δ). Lastly, we give a novel private algorithm that finds a depth parameter κ for which the volume of D_P(κ) is comparable to the volume of D_P(κ-Δ). We hope our work leads to the further study of the intersection of differential privacy and computational geometry.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol199-itc2021/LIPIcs.ITC.2021.18/LIPIcs.ITC.2021.18.pdf
Differential Privacy
Computational Geometry
Tukey Depth