arXiv：cs.LG· Youngjoo Yun, Rishabh Dudeja·· 5 小时前AI 评分32
差分隐私 PCA 的高维渐近理论分析
High-Dimensional Asymptotics of Differentially Private PCA
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研究针对含 n 个样本、p 个特征的差分隐私主成分分析（PCA），在高维极限（p→∞）下对指数机制给出了效用与隐私损失的精确渐近刻画。结果表明，在该极限下，利用隐私化主成分检测目标个体是否存在，渐近等价于区分两个均值不同的高斯分布，均值差取决于数据集的谱性质。分析结合了 Dong、Roth 和 Su（2022）提出的隐私保证假设检验框架与 Le Cam 的连续性论证。
正文
Abstract:In differential privacy, random noise is introduced to privatize summary statistics of a sensitive dataset before releasing them. The noise level determines the privacy loss, which quantifies how easily an adversary can detect a target individual's presence in the dataset using the published statistic. Most privacy analyses provide non-asymptotic upper bounds on the privacy loss which hold uniformly across all datasets. Sometimes, these bounds can be pessimistic on a given dataset. In such cases, it can be useful to complement these privacy bounds with sharp privacy characterizations that quantify a mechanism's exact privacy loss on a given dataset. With this goal, we study differentially private principal component analysis (PCA), where the goal is to privatize the leading principal components of a dataset with $n$ samples and $p$ features. We analyze the exponential mechanism and provide sharp asymptotic characterizations of its utility and privacy loss in the high-dimensional limit ($p \rightarrow \infty$). We show that in this limit, detecting a target individual's presence using privatized principal components is asymptotically equivalent to distinguishing between two Gaussians with different means, where the mean difference depends on certain spectral properties of the dataset. Our analysis combines the hypothesis-testing formulation of privacy guarantees proposed by Dong, Roth, and Su (2022) with Le Cam's contiguity arguments.
| Subjects: | Statistics Theory (math.ST); Information Theory (cs.IT); Machine Learning (cs.LG); Probability (math.PR); Machine Learning (stat.ML) |
| Cite as: | arXiv:2511.07270 [math.ST] |
| (or arXiv:2511.07270v4 [math.ST] for this version) | |
| https://doi.org/10.48550/arXiv.2511.07270 arXiv-issued DOI via DataCite |
Submission history
From: Youngjoo Yun [view email]
[v1]
Mon, 10 Nov 2025 16:17:16 UTC (3,809 KB)
[v2]
Sun, 15 Feb 2026 18:55:47 UTC (4,209 KB)
[v3]
Mon, 23 Feb 2026 03:48:49 UTC (4,220 KB)
[v4]
Fri, 2 Oct 2026 12:47:39 UTC (5,833 KB)
来源：arXiv：cs.LG · arxiv.org