  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.
