Privacy preservation has become a critical concern in high-dimensional data analysis due to the growing prevalence of data-driven applications. Proposed by Li (1991), sliced inverse regression has emerged as a widely utilized statistical technique for reducing covariate dimensionality while maintaining sufficient statistical information. In this paper, we propose optimally differentially private algorithms specifically designed to address privacy concerns in the context of sufficient dimension reduction. We proceed to establish lower bounds for differentially private sliced inverse regression in both the low and high-dimensional settings. Moreover, we develop differentially private algorithms that achieve the minimax lower bounds up to logarithmic factors. Through a combination of simulations and real data analysis, we illustrate the efficacy of these differentially private algorithms in safeguarding privacy while preserving vital information within the reduced dimension space. As a natural extension, we can readily offer analogous lower and upper bounds for differentially private sparse principal component analysis, a topic that may also be of potential interest to the statistical and machine learning community.

在高维数据分析中，隐私保护已成为一个关键问题，Li (1991)提出的切片逆回归技术作为一种广泛应用的统计技术，用于降低协变量维度并保持足够的统计信息。本文提出了专门针对充分维度减少情况下隐私问题的最优差分隐私算法，并建立了在低维和高维环境中差分隐私切片逆回归的下界。此外，我们开发了能够在对数因子范围内达到极小化界限的差分隐私算法。通过模拟和实际数据分析的结合，我们展示了这些差分隐私算法在保护隐私的同时在降维空间内保留重要信息的有效性。作为自然延伸，我们也可以提供差分隐私稀疏主成分分析的类似下界和上界，这也可能引起统计学和机器学习领域的兴趣。

差分隐私切片逆回归：极小极大优化和算法