标题： 鲁棒且差分隐私的PCA用于非高斯数据 显示英文标题

标题： Robust and Differentially Private PCA for non-Gaussian data

摘要： 最近的进展激发了对隐私保护主成分分析（PCA）发展的广泛关注。 然而，许多现有方法依赖于限制性假设，例如假设数据服从次高斯分布或容易受到数据污染的影响。 此外，一些方法计算成本高昂，或者依赖于必须估计的未知模型参数，这限制了寻求隐私保护PCA的数据分析师的使用。 在本文中，我们提出了一种适用于重尾和可能受污染数据的差分隐私PCA方法。 我们的方法利用了适当缩放数据的协方差矩阵在椭圆分布下保留特征向量及其顺序的性质，这些分布包括高斯分布和重尾分布。 通过应用有界变换，我们能够在差分隐私的情况下直接计算主成分。 此外，有界性保证了对数据污染的鲁棒性。 我们对所提出的方法进行了理论分析和实证评估，重点是其恢复由前导主成分张成子空间的能力。 广泛的数值实验表明，与现有方法相比，我们的方法在统计效用方面表现一致更好，特别是在非高斯或受污染的数据设置中。 显示英文摘要

摘要： Recent advances have sparked significant interest in the development of privacy-preserving Principal Component Analysis (PCA). However, many existing approaches rely on restrictive assumptions, such as assuming sub-Gaussian data or being vulnerable to data contamination. Additionally, some methods are computationally expensive or depend on unknown model parameters that must be estimated, limiting their accessibility for data analysts seeking privacy-preserving PCA. In this paper, we propose a differentially private PCA method applicable to heavy-tailed and potentially contaminated data. Our approach leverages the property that the covariance matrix of properly rescaled data preserves eigenvectors and their order under elliptical distributions, which include Gaussian and heavy-tailed distributions. By applying a bounded transformation, we enable straightforward computation of principal components in a differentially private manner. Additionally, boundedness guarantees robustness against data contamination. We conduct both theoretical analysis and empirical evaluations of the proposed method, focusing on its ability to recover the subspace spanned by the leading principal components. Extensive numerical experiments demonstrate that our method consistently outperforms existing approaches in terms of statistical utility, particularly in non-Gaussian or contaminated data settings.