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

Differentially Private Sliced Inverse Regression: Minimax Optimality and  Algorithm

Bayesian optimization (BO) has been broadly applied to computational
expensive problems, but it is still challenging to extend BO to high
dimensions. Existing works are usually under strict assumption of an additive
or a linear embedding structure for objective functions. This paper directly
introduces a supervised dimension reduction method, Sliced Inverse Regression
(SIR), to high dimensional Bayesian optimization, which could effectively learn
the intrinsic sub-structure of objective function during the optimization.
Furthermore, a kernel trick is developed to reduce computational complexity and
learn nonlinear subset of the unknowing function when applying SIR to extremely
high dimensional BO. We present several computational benefits and derive
theoretical regret bounds of our algorithm. Extensive experiments on synthetic
examples and two real applications demonstrate the superiority of our
algorithms for high dimensional Bayesian optimization.

本文提出了一种基于 Sliced Inverse Regression（SIR）的高维贝叶斯优化算法，能够通过降维方法学习对象函数的内在子结构，并且采用了核技巧来降低计算复杂度和学习非线性的子集。实验证明，该算法在高维贝叶斯优化中具有优越性。

高维贝叶斯优化之监督降维

High Dimensional Bayesian Optimization via Supervised Dimension  Reduction

It is known that for a certain class of single index models (SIMs) $Y =
f(\boldsymbol{X}_{p \times 1}^\intercal\boldsymbol{\beta}_0, \varepsilon)$,
support recovery is impossible when $\boldsymbol{X} \sim \mathcal{N}(0,
\mathbb{I}_{p \times p})$ and a model complexity adjusted sample size is below
a critical threshold. Recently, optimal algorithms based on Sliced Inverse
Regression (SIR) were suggested. These algorithms work provably under the
assumption that the design $\boldsymbol{X}$ comes from an i.i.d. Gaussian
distribution. In the present paper we analyze algorithms based on covariance
screening and least squares with $L_1$ penalization (i.e. LASSO) and
demonstrate that they can also enjoy optimal (up to a scalar) rescaled sample
size in terms of support recovery, albeit under slightly different assumptions
on $f$ and $\varepsilon$ compared to the SIR based algorithms. Furthermore, we
show more generally, that LASSO succeeds in recovering the signed support of
$\boldsymbol{\beta}_0$ if $\boldsymbol{X} \sim \mathcal{N}(0,
\boldsymbol{\Sigma})$, and the covariance $\boldsymbol{\Sigma}$ satisfies the
irrepresentable condition. Our work extends existing results on the support
recovery of LASSO for the linear model, to a more general class of SIMs.

本研究针对一类单指数模型，在不同的数据分布和假设条件下分别采用切片逆回归、LASSO 以及协方差筛选等算法，实现对支持集的恢复，并对 LASSO 用于线性模型的支持恢复结果进行推广。