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arXiv 2510.26679cs.LGcs.DS
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基于矩阵一致性的紧差分私有PCA

Tight Differentially Private PCA via Matrix Coherence

Bocconi University(博科尼大学);Lucerne School of Computer Science and Information Technology(卢塞恩计算机科学与信息科技学校)

Tommaso d'Orsi, Gleb Novikov

摘要:

我们重新审视的任务,计算跨度的顶部$r$奇异向量$u_1, \ldots, u_r$的矩阵下差分隐私。我们表明,一个简单而有效的算法-基于奇异值分解和标准的扰动机制-返回一个私人的秩$r$近似,其误差仅取决于\emph{rank-$r$ coherence}的$u_1, \ldots, u_r$和频谱间隙$σ_r - σ_{r+1}$。这解决了Hardt和Roth~ \cite{hardt2013beyond}提出的一个问题。我们的估计器优于最先进的技术-在某些情况下显着。特别是,我们表明,在密集的设置,它实现了相同的保证单尖峰PCA的Wishart模型中的最佳非私人算法所达到的,而以前的私人算法未能做到这一点。此外,我们证明了(秩- $r$)相干性不增加高斯扰动下。这意味着任何基于高斯机制的估计器-包括我们的估计器-都保持了输入的一致性。我们推测,类似的行为适用于其他结构化模型,包括种植的问题图。我们还探讨了应用程序的连贯性图的问题。特别是,我们提出了一个差分隐私算法的最大切割和其他约束满足问题下的低一致性假设。

英文摘要:

We revisit the task of computing the span of the top $r$ singular vectors $u_1, \ldots, u_r$ of a matrix under differential privacy. We show that a simple and efficient algorithm -- based on singular value decomposition and standard perturbation mechanisms -- returns a private rank-$r$ approximation whose error depends only on the \emph{rank-$r$ coherence} of $u_1, \ldots, u_r$ and the spectral gap $σ_r - σ_{r+1}$. This resolves a question posed by Hardt and Roth~\cite{hardt2013beyond}. Our estimator outperforms the state of the art -- significantly so in some regimes. In particular, we show that in the dense setting, it achieves the same guarantees for single-spike PCA in the Wishart model as those attained by optimal non-private algorithms, whereas prior private algorithms failed to do so. In addition, we prove that (rank-$r$) coherence does not increase under Gaussian perturbations. This implies that any estimator based on the Gaussian mechanism -- including ours -- preserves the coherence of the input. We conjecture that similar behavior holds for other structured models, including planted problems in graphs. We also explore applications of coherence to graph problems. In particular, we present a differentially private algorithm for Max-Cut and other constraint satisfaction problems under low coherence assumptions.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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