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Differentially Private Approximations of a Convex Hull in Low Dimensions

Authors: Yue Gao and Or Sheffet

Published in: LIPIcs, Volume 199, 2nd Conference on Information-Theoretic Cryptography (ITC 2021)

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.

Cite as

Yue Gao and Or Sheffet. Differentially Private Approximations of a Convex Hull in Low Dimensions. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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  author =	{Gao, Yue and Sheffet, Or},
  title =	{{Differentially Private Approximations of a Convex Hull in Low Dimensions}},
  booktitle =	{2nd Conference on Information-Theoretic Cryptography (ITC 2021)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-197-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{199},
  editor =	{Tessaro, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-143377},
  doi =		{10.4230/LIPIcs.ITC.2021.18},
  annote =	{Keywords: Differential Privacy, Computational Geometry, Tukey Depth}
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