提出可同时处理有监督与无监督数据的非线性降维方法，兼具可解释性与高效性。
Nonlinear Multiple Response Regression and Learning of Latent Spaces
- 基于指数模型的非线性多响应回归框架，用广义Stein引理估计隐空间。
- 无需知道非线性函数形式即可学习隐变量，理论支持强。
- 比自编码器更可解释、计算更快，适合需要透明性的实际应用。
在高维数据中识别低维潜在结构是机器学习的核心课题，旨在实现数据压缩、存储、传输与深层理解。传统方法如主成分分析（PCA）和自编码器（AE）均为无监督，忽略已有标签信息。本文提出一种统一方法，可在无监督与有监督设置下学习隐空间。将问题建模为指数模型中的非线性多响应回归，利用广义Stein引理，在无需已知非线性链接函数的情况下估计隐空间。该方法可视为PCA的非线性推广。与自编码器等神经网络方法作为“黑箱”不同，本方法具有更好可解释性，降低计算复杂度，并提供强理论保障。大量数值实验与真实数据分析验证了其优越性能。
原文摘要 · Abstract (English)
Identifying low-dimensional latent structures within high-dimensional data has long been a central topic in the machine learning community, driven by the need for data compression, storage, transmission, and deeper data understanding. Traditional methods, such as principal component analysis (PCA) and autoencoders (AE), operate in an unsupervised manner, ignoring label information even when it is available. In this work, we introduce a unified method capable of learning latent spaces in both unsupervised and supervised settings. We formulate the problem as a nonlinear multiple-response regression within an index model context. By applying the generalized Stein's lemma, the latent space can be estimated without knowing the nonlinear link functions. Our method can be viewed as a nonlinear generalization of PCA. Moreover, unlike AE and other neural network methods that operate as "black boxes", our approach not only offers better interpretability but also reduces computational complexity while providing strong theoretical guarantees. Comprehensive numerical experiments and real data analyses demonstrate the superior performance of our method.
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