低维空间中凸包的隐私近似
Private Approximations of a Convex Hull in Low Dimensions
AI总结：
该研究提出了一种在低维空间中具有差分隐私的算法，用于近似凸包的几何特征，并通过双标准近似方法生成方向核。
摘要：
我们给出了首个具有差分隐私的算法，用于估计欧几里得空间中点集的各种几何特征，如直径、宽度、凸包体积、最小包围盒、最小包围球等。我们的工作严重依赖于Tukey深度的概念。不同于（非隐私地）近似给定点集P的凸包，我们的算法近似由P诱导的κ-Tukey区域的几何特征（所有Tukey深度为κ或更大的点）。此外，我们的近似都是双标准的：对于任何几何特征μ，我们的(α,Δ)-近似值介于(1-α)μ(D_P(κ))和(1+α)μ(D_P(κ-Δ))之间。我们的工作旨在产生一个(α,Δ)-核D_P(κ)，即一个集合S，使得（在平移后）有(1-α)D_P(κ)⊂CH(S)⊂(1+α)D_P(κ-Δ)。我们证明了类似于Agarwal等人[2004]最初提出的方向核双标准近似概念会失败，无法产生核，因此我们转向更细致的投影近似概念，这些近似确实能产生核。首先，我们给出了寻找“肥”Tukey区域(α,Δ)-核的差分隐私算法。然后，基于对最小包围盒的私有近似，我们找到一个转换，将D_P(κ)转换为“肥”区域，但仅当其体积与D_P(κ-Δ)的体积成比例时。最后，我们给出了一个新颖的私有算法，找到一个深度参数κ，使得D_P(κ)的体积与D_P(κ-Δ)的体积可比。我们希望这项工作能促进差分隐私与计算几何交集的进一步研究。
英文摘要：
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 \emph{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 $κ$-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 \emph{$(α,Δ)$-kernel of $D_P(κ)$}, namely a set $\mathcal{S}$ such that (after a shift) it holds that $(1-α)D_P(κ)\subset \mathsf{CH}(\mathcal{S}) \subset (1+α)D_P(κ-Δ)$. We show that an analogous notion of a bi-critera approximation of a directional kernel, as originally proposed by Agarwal et al~[2004], \emph{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 \emph{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 $D_P(κ-Δ)$. We hope this work leads to the further study of the intersection of differential privacy and computational geometry.
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