In this work, we develop a new theory and method for sufficient dimension
reduction (SDR) in single-index models, where SDR is a sub-field of supervised
dimension reduction based on conditional independence. Our work is primarily
motivated by the recent introduction of the Hellinger correlation as a
dependency measure. Utilizing this measure, we develop a method capable of
effectively detecting the dimension reduction subspace, complete with
theoretical justification. Through extensive numerical experiments, we
demonstrate that our proposed method significantly enhances and outperforms
existing SDR methods. This improvement is largely attributed to our proposed
method's deeper understanding of data dependencies and the refinement of
existing SDR techniques.

本文提出了一种新的理论和方法，用于单指数模型中的充分降维 (SDR)，SDR 是基于条件独立性的监督降维的一个子领域。利用 Hellinger 相关作为一种依赖度量，我们开发了一种能够有效检测降维子空间的方法，并给出了理论解释。通过大量的数值实验，我们证明了我们提出的方法明显优于现有的 SDR 方法。这种改进主要归因于我们提出的方法对数据依赖性的更深入理解和对现有 SDR 技术的改进。

通过 Hellinger 相关性增强充分降维

Enhancing Sufficient Dimension Reduction via Hellinger Correlation

Recent works investigated the generalization properties in deep neural
networks (DNNs) by studying the Information Bottleneck in DNNs. However, the
mea- surement of the mutual information (MI) is often inaccurate due to the
density estimation. To address this issue, we propose to measure the dependency
instead of MI between layers in DNNs. Specifically, we propose to use
Hilbert-Schmidt Independence Criterion (HSIC) as the dependency measure, which
can measure the dependence of two random variables without estimating
probability densities. Moreover, HSIC is a special case of the Squared-loss
Mutual Information (SMI). In the experiment, we empirically evaluate the
generalization property using HSIC in both the reconstruction and prediction
auto-encoding (AE) architectures.

研究通过使用 Hilbert-Schmidt 独立准则（HSIC）作为依赖度量来测量 DNN 中图层之间的依赖关系，从而更准确地研究 DNN 中的信息瓶颈和泛化特性，并在实验中评估了该方法在 auto-encoding 的应用。