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 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.
Differential Privacy
Computational Geometry
Tukey Depth
Theory of computation~Theory of database privacy and security
18:1-18:16
Regular Paper
Much of this work was done when the first author was adviced by the second author, and was supported by grant #2017–06701 of the Natural Sciences and Engineering Research Council of Canada (NSERC). In addition, this work was partially done when O.S. was a participant of the Simons' Institute for the Theory of Computing program of Data-Privacy. O.S. is supported by the BIU Center for Research in Applied Cryptography and Cyber Security in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Office, and by ISF grant no. 2559/20.
http://arxiv.org/abs/2007.08110
Both authors thank the anonymous reviewers for many helpful suggestions in improving this paper.
Yue
Gao
Yue Gao
Department of Computing Science, University of Alberta, Edmonton, Canada
Or
Sheffet
Or Sheffet
Faculty of Engineering, Bar-Ilan University, Ramat Gan, Israel
https://orcid.org/0000-0002-5182-0530
10.4230/LIPIcs.ITC.2021.18
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Yue Gao and Or Sheffet
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