Intrinsic-Dimensional Wasserstein Guarantees for Private Synthetic Measures

2026-09-15

AI Summary

本文研究了在最坏情况数据模型下，使用PrivTree算法构建自适应二叉分区并发布其叶质量以生成ε-差分隐私合成度量的方法，该方法的1-Wasserstein误差依赖于内在维度而非环境维度。

Abstract

We study an $\varepsilon$-differentially private synthetic measure for $n$ points in $[0,1]^d$ by applying the existing PrivTree algorithm to construct an adaptive binary partition and then privately releasing its leaf masses. We consider the worst-case data model without any sampling or population-distribution assumption. The 1-Wasserstein error of the synthetic measure is $\widetilde O_d((\varepsilon n)^{-1/d})$ for $d\ge2$, which is optimal compared to the minimax lower bound up to a logarithmic factor. Moreover, for $d\ge3$ and $2<s\le d$, if the data set has covering number at most $Ar^{-s}$ over the relevant finite range of scales $r$, the expected error improves to $\widetilde O_{d,s}((\varepsilon n)^{-1/s})$. Thus the rate depends on a finite-scale intrinsic dimension rather than the ambient dimension, without requiring the recovery of a low-dimensional manifold. We also introduce a shifting technique to further avoid the exponential dependence of the constant on the ambient dimension $d$.

Problem

• differential privacy
• synthetic measure
• Wasserstein error
• intrinsic dimension

Innovation

• ε-differentially private
• synthetic measure
• intrinsic dimension
• Wasserstein error
• shifting technique